Mathematical Biology
J. Math. Biol. DOI 10.1007/s00285-013-0742-y
The Allee-type ideal free distribution Vlastimil Kˇrivan
Received: 22 May 2013 / Revised: 20 November 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space. Keywords Dispersal · Evolutionary game theory · Habitat selection game · Logistic equation · Non-smooth analysis · Population dynamics · Optimal foraging Mathematics Subject Classification
26A27 · 92D25 · 92D40 · 92D50
1 Introduction Animal distribution in fragmented landscapes is often described by the ideal free distribution (Fretwell and Lucas 1969; Fretwell 1972). This theory predicts that animals distribute so that payoffs in all occupied patches are the same and maximal. When patch payoffs are negatively density dependent the IFD is an evolutionarily stable strategy of the habitat selection game (Kˇrivan et al. 2008). The habitat selection game
V. Kˇrivan (B) Biology Center, Academy of Sciences of the Czech Republic, and Faculty of Science, University of South Bohemia, Branišovská 31, 370 05 Ceske Budejovice, Czech Republic e-mail: [email protected]
123
V. Kˇrivan
is a game theoretical concept (similar to the Hawk–Dove game or to the Prisoner’s dilemma) that seeks evolutionarily stable animal distributions in space. In their pioneering article Fretwell and Lucas (1969) also briefly mentioned the Allee-type IFD where patch payoffs are positively density dependent at low population densities. In particular, using graphical analysis they predicted that a small change in population abundance can lead to an abrupt (discontinuous) change in population distribution. For example, when all individuals occupy patch 1 at low population densities there can be a population threshold at which population distribution changes discontinuously. Morris (2002) used the Allee-type IFD to study distributions of the deer mice. While the Fretwell and Lucas’ geometric analysis assumed a general hump-shaped patch payoff function, Morris (2002) considered payoffs that first linearly increase and then linearly decrease with population abundance. Such piece-wise linear payoff functions qualitatively capture properties of many Allee-type fitness functions used in the literature (Courchamp et al. 2008), and they are easier to analyze when compared with non-linear functions. Both Fretwell and Lucas (1969) and Morris (2002) assumed a particular case where the payoff in one patch is always higher than the payoff in the other patch. In other words, the graphs of these two payoffs never intersected (e.g., see Figure 2 in Morris 2002). Fretwell and Lucas (1969) concluded by “One can imagine [payoff] curves which lead to complete shifts in population [distribution] while other curves may lead to no erratic behavior at all”. In this article I will analyze such more general patch payoffs and will provide conditions under which erratic changes in distributions do occur. I will show that for some intermediate population densities there are up to three possible IFDs and there are three alternative population equilibria. I will also study population and distributional dynamics under the assumption that individuals disperse in the direction of increasing fitness. 2 Model I consider a single population in a heterogeneous environment consisting of two patches. Following Morris (2002) I assume piecewise linear patch payoffs (Fig. 1a) xi , i = 1, 2, Vi (xi ) = min ai (xi − Ai ) , ri 1 − Ki where xi is the population abundance in patch i, ri is the intrinsic per capita population growth rate, K i is the patch carrying capacity, Ai is the Allee threshold, and ai is the strength of positive density dependence at low population densities. I will assume that, except the Allee threshold, all parameters are positive. The Allee threshold is the population density below which the population growth rate is negative and above which it is positive. If the Allee threshold is positive, the Allee effect is strong, because at low population densities the per capita population growth rate is negative and the population dies out. If the Allee threshold is negative, then the Allee effect is weak because even at low population densities the per capita population growth rate is positive. Function Vi has one maximum at population density ximax =
123
K i (ai Ai + ri ) , i = 1, 2 ai K i + ri
The Allee-type ideal free distribution
at which the patch payoff is Vi (ximax ) =
ai ri (K i − Ai ) , i = 1, 2. ai K i + ri
I will assume throughout the article that ximax > 0 (i = 1, 2) which means ai Ai + ri > 0,
i = 1, 2.
(1)
If the opposite inequality holds, patch payoffs would be negatively density dependent for all population densities (i.e., no Allee effect). In addition, I will assume that Vi (ximax ) > 0 (i.e., K i > Ai ) so that the payoff functions have a triangular form (Fig. 1a–c) qualitatively similar to other models of the per capita population growth rate with the Allee effect used in the literature (see Table 3.1 on p. 68 in Courchamp et al. 2008). In what follows I am interested in the distribution (x1 , x2 ) of total x = x1 + x2 individuals among the two habitats. If u i denotes the proportion of the lifetime an average individual spends in patch i = 1, 2 and the population is monomorphic then xi = u i x. Thus, the proportion of time spend in each of the two patches (u 1 , u 2 ) (u 1 + u 2 = 1) is the individual strategy. Fitness of a mutant with strategy u˜ = (u˜ 1 , u˜ 2 ) in the resident population with strategy u = (u 1 , u 2 ) is calculated as the average payoff W (u, ˜ u; x) = u˜ 1 V1 (u 1 x) + u˜ 2 V2 (u 2 x). The underlying game was called the habitat selection game (HSG; Kˇrivan et al. 2008; Kˇrivan and Cressman 2009; Cressman and Kˇrivan 2010). The Nash equilibria of the HSG coincide with the ideal free distribution concept [IFD; Fretwell and Lucas (1969); Kˇrivan et al. (2008)]. When patch payoffs are negatively density dependent, there is a unique Nash equilibrium of the HSG which is also an evolutionarily stable strategy (Cressman and Kˇrivan 2006; Kˇrivan et al. 2008). In this article I will study evolutionarily stable strategies (called here IFDs) (u ∗1 , u ∗2 ) for fitness function W. Because the fitness function is non-linear in the resident strategy u = (u 1 , u 2 ), I use the local evolutionary stability condition (Pohley and Thomas 1983; Hofbauer and Sigmund 1998) that classifies strategy u ∗ = (u ∗1 , u ∗2 ) as an (local) ESS if W (u ∗ , u; x) > W (u, u; x) for all strategies u = (u 1 , u 2 ) = (u ∗1 , u ∗2 ) = u ∗ in a neighborhood of u ∗ . On the other hand, since W (u ∗ , u; x) is linear in u ∗1 and u ∗2 , patch payoffs must be equal at an ESS (i.e. V1 (u ∗1 x) = V2 (u ∗2 x)) if 0 < u ∗1 < 1 (Hofbauer and Sigmund 1998). In what follows I will need the following threshold total patch-wide population abundances: x˜ 1 =
a1 K 2 r1 (A1 − K 1 ) + a1 K 1r2 (A1 + K 2 ) + r1r2 (K 1 + K 2 ) , r2 (a1 K 1 + r1 )
123
123
30
40
50
0
10
20
30
0
10
20
30 40 50
H
10
0
10
20
0
0
30
0.0
0.0
20
0.2
1.0
0.2
E
0.4
40
0.4
30
0.6
20
0.6
10
10
0.5
0.0
0.5
1.0
1.5
2.0
0.8
0
0
B
0.8
1.0
2
1
0
1
2
0
10
10
5
20
20
10
30
30
15
40
40
20
50
50
25
I
F
C
0
50
0
50
0
0
100
0.0
0.2
0.4
0.6
0.8
1.0
1
0
1
2
10
10
5
20
20
10
30
30
15
40
40
20
50
50
25
60
60
30
70
70
35
Fig. 1 a–c Payoffs V1 (solid line) and V2 (dashed line) as a function of x1 and x2 , respectively. d–f Show corresponding IFDs (u 1 ; here segments u i1 = x1i /x, i = 1, . . . , 5 correspond to distributions (a)–(e) defined in the text) as a function of total population abundance x. g–i The population growth rate x1 V1 (x1 ) + x2 V2 (x2 ) under the IFD. The black dots denote locally stable population equilibria, while light gray dots are unstable population equilibria. The line styles in g–i corresponds to those in d–f. Parameters used in a, d and g: r1 = 3.5, r2 = 2.5, a1 = 0.25, a2 = 0.3, K 1 = 20, K 2 = 25, A1 = 1.5, A2 = 3.5. Parameters used in b, e and h: r1 = 3.5, r2 = 2., K 1 = 20, K 2 = 25, A1 = 1.5, A2 = 3.5, a1 = 0.25, a2 = 0.15, Parameters used in c, f and i: r1 = 5, r2 = 10, K 1 = 35, K 2 = 30, A1 = 9, A2 = 2, a1 = 0.15, a2 = 0.1
G
D
A
V. Kˇrivan
The Allee-type ideal free distribution
a2 r2 (K 2 − A2 ) + a1r2 (A1 + K 2 ) + a1 a2 K 2 (A1 + A2 ) , a1 (a2 K 2 + r2 ) a2 K 2 r1 (A2 + K 1 ) + a2 r2 K 1 (A2 − K 2 ) + r1r2 (K 1 + K 2 ) x˜ 3 = , r1 (a2 K 2 + r2 ) a1 a2 K 1 (A1 + A2 ) + a1 r1 (K 1 − A1 ) + a2 r1 (A2 + K 1 ) . x˜ 4 = a2 (a1 K 1 + r1 )
x˜ 2 =
(2)
These abundances correspond to population distributions at which patch payoffs are the same (i.e., V1 (x1 ) = V2 (x2 )) and the patch payoff at least in one patch is maximal (i.e., the population density in at least one patch equals ximax ). In particular x˜ 1 assumes that V2 (x2max ) exceeds V1 (x1max ). In this case, the equation V1 (x1max ) = V2 (x2 ) always has (A1 −K 1 )K 2 r1 solution x2 = K 2 + a1(a which is in the range of population densities at which 1 K 1 +r1 )r2 the patch 2 payoff is negatively density dependent. For example, in Fig. 1c, x1max = 21.7 and x2 = 24.3 so that x˜ 1 = 46.0. When V2 (0) < V1 (x1max ), equation V1 (x1max ) = 1 −A1 ) V2 (x2 ) has another solution x2 = A2 + aa12r(a1 (K (e.g., x2 = 21.0 in Fig. 1c) 1 K 1 +r1 ) which is in the range of population densities where the patch 2 payoff is positively density dependent. This population distribution (x1max , x2 ) defines the total population threshold x˜ 4 . Similarly, x˜ 3 assumes that V1 (x1max ) exceeds V2 (x2max ) (Fig. 1a, b). In this 2 r2 (A2 −K 2 ) case, the equation V1 (x1 ) = V2 (x2max ) always has solution x1 = K 1 + K r11a(a 2 K 2 +r2 ) which is in the range of population densities at which the patch 1 payoff is negatively density dependent. When V1 (0) < V2 (x2max ), equation V1 (x1 ) = V2 (x2max ) has another 2 −A2 ) solution x1 = A1 + aa21r(a2 (K which is in the range of population densities at 2 K 2 +r2 ) which the patch 1 payoff is positively density dependent. This population distribution (x1 , x2max ) defines the total population threshold x˜ 2 . For example, in Fig. 1b, x1 = 5.9 and x2max = 10.9 so that x˜ 2 = 16.8. Appendix A shows that there are the following IFDs: (a) For population densities that satisfy (r1 + a2 A2 )K 1 a1 A 1 − a2 A 2 ≤x≤ a1 r1 distribution (x11 , x21 ) = (x, 0) where all individuals occupy patch 1 is an IFD. E.g., when patch 1 is more profitable than patch 2 when both patches are unoccupied, i.e., V1 (0) = −a1 A1 > V2 (0) = −a2 A2 , (1) implies that r1 + a2 A2 > 0 and the above inequality holds for small population densities. (b) For population densities that satisfy (r2 + a1 A1 )K 2 a2 A 2 − a1 A 1 ≤x≤ a2 r2 distribution (x12 , x22 ) = (0, x) is an IFD and all individuals occupy patch 2 only. (c) For population densities that satisfy K 2 (a1 A1 + r2 ) < x < min{x˜ 1 , x˜ 2 }, r2
123
V. Kˇrivan
distribution (x13 , x23 )
=
r2 (x − K 2 ) − a1 A1 K 2 K 2 (r2 + a1 (A1 − x)) , r 2 − a1 K 2 r 2 − a1 K 2
(3)
is an IFD provided a1 < r2 /K 2 . At this distribution density dependence in patch 1 is positive and density dependence in patch 2 is negative (i.e., 0 < x13 ≤ x1max and x2max ≤ x23 < x). (d) For population densities that satisfy K 1 (a2 A2 + r1 ) < x < min{x˜ 3 , x˜ 4 }, r1 distribution (x14 , x24 )
=
K 1 (r1 + a2 (A2 − x)) r1 (x − K 1 ) − a2 A2 K 1 , r 1 − a2 K 1 r 1 − a2 K 1
(4)
is an IFD provided a2 < r1 /K 1 . At this distribution density dependence in patch 1 is negative and density dependence in patch 2 is positive (i.e., x1max ≤ x14 and 0 < x24 ≤ x2max ). (e) For population densities that satisfy max{x˜ 1 , x˜ 3 } < x, distribution (x15 , x25 ) =
K 1 (K 2 (r1 − r2 ) + r2 x) K 2 (K 1 (r2 − r1 ) + r1 x) , K 2 r1 + K 1 r2 K 2 r1 + K 1 r2
(5)
is an IFD. At this equilibrium density dependence is negative at both patches (i.e., x1max < x15 and x2max < x25 ). A. It The case where either x1 = x1max or x2 = x2max is discussed in Appendix a1 r1 K 2 (A1 −K 1 ) max is an IFD is shown there that population distribution x1 , K 2 + (a1 K 1 +r1 )r2 −K 2 ) max is an if a1 < r2 /K 2 . Similarly, population distribution K 1 + a2rr12(aK21 K(A22+r , x 2 2) IFD provided a2 < r1 /K 1 . Population distribution (x1max , x2max ) is an IFD provided a1 < r2 /K 2 and a2 < r1 /K 1 . Dependence of these IFDs on population abundance is shown in Fig. 1d–f, where u i1 = x1i /x, i = 1, . . . , 5. For low population densities all individuals occupy the patch that has the higher payoff when unoccupied. It is patch 1 in Fig. 1d, e, where V1 (0) > V2 (0) and patch 2 in Fig. 1f where V1 (0) < V2 (0). Because at low population densities patch payoffs are positively density dependent, individuals prefer to stay together in the more profitable patch as population abundance increases. Due to initial positive density dependence of patch payoffs there is also an alternative distribution at slightly higher population abundances under which all individuals occupy the patch
123
The Allee-type ideal free distribution
that was initially inferior (this is patch 2 in Fig. 1d, e and patch 1 in Fig. 1f). The minimal population density for which this alternative distribution is possible satisfies V2 (x) = V1 (0) in Fig. 1d and e, and V1 (x) = V2 (0) in Fig. 1f. This is because when the originally poorer patch is occupied by a population that is abundant enough, its patch payoff will be higher than in the unoccupied patch. As the overall population density increases, the payoff in the occupied patch reaches its maximum and starts to decline due to negative density dependence. When the decreasing payoff in the occupied patch reaches the value that equals the payoff in the unoccupied patch there are two qualitatively different possibilities. Let us consider the case where all individuals occupy patch 1 in Fig. 1d (the solid line). At the critical population density x = 26, V1 (x) = V2 (0) (Fig. 1a). I remark, that in this case population distributions (x13 , x23 ) and (x14 , x24 ) do not exist because K 2 (a1 A1 + r2 )/r2 = 28.75 is greater than is min{x˜ 1 , x˜ 2 } = min{15.07, 16.82} and K 1 (a2 A2 + r1 )/r1 = 26 is greater than min{x˜ 3 , x˜ 4 } = min{19.66, 18.96}. Thus, the only possible distribution for x ≥ 26 is the interior distribution (x15 , x25 ) (the short dashed line denoted by u 51 in Fig. 1d). Because for x = 26 this distribution is (x15 , x25 ) = (13, 13) there is a discontinuity in Fig. 1d at x = 26. Now I will consider the situation shown in Fig. 1e where at the critical population density x = 23 distribution at which all individuals occupy patch 1 only changes continuously to a distribution where both patches are occupied. This is because in this plot parameters are such that K 1 (a2 A2 + r1 )/r1 = 23 < min{x˜ 3 , x˜ 4 } = min{24.5, 25.3} and a2 = 0.15 < r1 /K 1 = 0.175. Thus, as population density increases above 23, the population will continuously re-distribute over both patches following distribution (x14 , x24 ). Under this distribution the payoffs in both patches will increase until population density in patch 2 will reach x2max . This happens when the overall population density equals x˜ 3 = 24.5. From then on, payoffs in both patches will decrease and the distribution will continuously change according to distribution (x15 , x25 ). The case shown in Fig. 1f where all individuals occupy patch 1 only is similar, but here distribution (x14 , x24 ) changes discontinuously to (x13 , x23 ). This is because payoffs in both patches increase for increasing population densities until the population density in patch 1 reaches x1max . At this moment the corresponding distribution in the second patch is still in the region where patch 2 is positively density dependent. Consequently, this distribution cannot be continuously extended for higher population densities. 3 Population dynamics The overall population dynamics are described by the following differential equation dx = u 1 x V1 (u 1 x) + u 2 x V2 (u 2 x) = x V dt
(6)
where (u 1 , u 2 ) denotes the population distribution among the two patches. This population model is an extension of a population growth with the Allee effect to two-patch environments. When u 1 and u 2 are fixed, i.e., when individual strategy is non-adaptive, the population growth rate is shown in Fig. 2. Provided the Allee effect is strong
123
V. Kˇrivan
30
20
10
0 0
10
20
30
40
50
Fig. 2 Population growth rate (x1 V1 (x1 ) + x2 V2 (x2 )) in model (6) for a fixed population distribution (u 1 , u 2 ) = (0.6, 0.4). Other parameters are the same as those used in Fig. 1a
(u 1 V1 (0) + u 2 V2 (0) = −u 1 a1 A1 − u 2 a2 A2 < 0), for low initial population densities population dynamics converge to extinction, while above the Allee threshold population dynamics tend to a positive equilibrium. Here I consider the situation where the population distributes very fast according to the IFD derived in the previous section. In other words, dispersal dynamics are assumed to be (infinitely) fast when compared with population dynamics. From the previous section it follows that there are up to three IFDs at a given positive population density. Figure 1g–i shows the mean population growth rate (x V ) for each possible IFD (shown in Fig. 1d–f; the line type correspond in these panels). Only those equilibria where the population growth rate is negatively density dependent are locally asymptotically stable (black dots). Thus, there are up to three locally stable positive population equilibria. If all individuals are in patch 1 (patch 2) only, the corresponding positive equilibrium is K 1 (K 2 ). When the population occupies both habitats, the only positive population equilibrium is K 1 + K 2 which corresponds to the IFD where both patches are occupied up to their carrying capacities. It is interesting to note that population equilibria that correspond to IFDs at which the payoff in one patch increases and decreases in the other patch (these IFDs are given by (3) or (4) and correspond to the two middle length dashed curves in Fig. 1e, f) can never be stable, because the per capita population growth rate at these distributions is always positively density dependent (see Fig. 1h, i where corresponding unstable equilibria are x ∼ 23 in Fig. 1h and x ∼ 37 and x ∼ 39 in Fig. 1i). This shows that distributional stability is different from population stability. In other words, population dynamics can destabilize distributions that are evolutionarily stable at a fixed population density. When population and distributional time scales cannot be separated, i.e., when both processes operate on similar time-scales, distributional dynamics can be described by the replicator equation (Taylor and Jonker 1978; Hofbauer and Sigmund 1998; Tran and Cressman 2014) du 1 = δu 1 (1 − u 1 )(V1 (u 1 x) − V2 (u 2 x)) dt
123
(7)
The Allee-type ideal free distribution
where u 2 = 1 − u 1 , and δ > 0 shows how demographic and distributional time scales are related (Cressman and Krivan 2013). This equation together with population dynamics (6) describes linked changes in population dynamics and distribution. It is proved in Appendix C that the extinction equilibrium (x = 0) is locally asymptotically stable when V1 (0) + V2 (0) = −a1 A1 − a2 A2 < 0. When V1 (0) = −a1 A1 < 0 (V2 (0) = −a2 A2 < 0) then the equilibrium at which all individuals occupy patch 2 (patch 1) only is locally asymptotically stable. Finally, the interior equilibrium where x1 = K 1 , x2 = K 2 , u 1 = K 1 /(K 1 + K 2 ), and u 2 = K 2 /(K 1 + K 2 ) is always locally asymptotically stable. These predictions do not differ from those where I assume that the population distribution is at the IFD at the current population density, i.e., when distributional dynamics are infinitely faster when compared with population dynamics. However, for more complex models population stability can depend on time scales (Cressman and Krivan 2013).
4 Discussion In this article I have studied the Allee-type ideal free distribution. I have shown that depending on parameters and population densities the Allee effect leads to multiple IFDs. They correspond to distributions where all individuals occupy patch 1 only, patch 2 only, or both patches. When population dynamics are considered, there can be up to three locally stable population equilibria. I have also shown that evolutionary stability of a population distribution does not guarantee population stability, because a population equilibrium which is also an IFD does not need to be locally stable. The IFD was introduced by Fretwell and Lucas (1969) who defined it as a Nash equilibrium. When density dependence is negative, it was shown (Cressman and Kˇrivan 2006; Kˇrivan et al. 2008) that there is a unique IFD that is also an evolutionarily stable strategy of the underlying habitat selection game. This means that this unique IFD is resistant to small perturbations. Fretwell and Lucas (1969) also considered the Allee-type IFD where density dependence is positive at low population densities. They showed that the IFD can dramatically (discontinuously) change at some critical overall population densities. They concluded that “...species following Allee’s principle may demonstrate erratic changes in distribution with small changes in population.” Their work was continued by Morris (2002) who analyzed such erratic changes in distribution of the deer mice. Both these works assumed patch payoffs without an intersection and the analysis was purely graphical. While Fretwell and Lucas (1969) assumed smooth patch payoffs, Morris (2002) assumed piece-wise linear patch payoffs. Such piecewise linear patch payoffs qualitatively capture properties of many hump shaped fitness functions used in the literature on the Allee effect (Courchamp et al. 2008). The advantage of piece-wise linear functions when compared with non-linear functions used in the literature on the Allee effect is that they are more readily analyzable. In this article, following Morris (2002), I have analyzed the general case where patch payoffs are piece-wise linear and they can intersect. Depending on model parameters and overall population density I have analyzed all corresponding IFDs. I have shown that up to three IFDs can co-exist at a single population abundance. However, this can happen only for intermediate population densities where the Allee effect influences
123
V. Kˇrivan Fig. 3 Isodars corresponding to IFDs shown in Fig. 1d–f. Isodars connect population numbers in patch 1 and patch 2 phase space along which the expected fitness is the same in both patches (i.e., V1 (x1 ) = V2 (x2 )). The lines along the coordinate axes correspond to distributions where the population occupies one patch only. When population numbers are low, the interior isodar does not exists (a) because all individuals occupy patch 1 or patch 2 only due to the Allee effect. Depending on patch payoffs, the interior isodar can be piece-wise linear (b, c). The line types in these panels correspond to line types in Fig. 1d–f. Parameters are the same as those used in Fig. 1
A 30 25 20 15 10 5 0 0
5
10
15
20
25
30
0
5
10
15
20
25
30
B 40
30
20
10
0
C 40
30
20
10
0 0
10
20
30
40
the population distribution at least in one patch. At high population densities where both patch payoffs are negatively density dependent there is a unique IFD (Fig. 1d–f). At intermediate population densities two pure strategies where all individuals occupy patch 1 or patch 2 only exist. At very low population densities only the patch with the highest payoff when unoccupied is populated. Whether there will be discontinuous
123
The Allee-type ideal free distribution
(erratic) changes in the IFD as overall population density increases depends on the history and slopes of the two payoff functions. For example, in Fig. 1a and d, as population size increases, there will be a sudden change at the IFD. However, this is not always so because the population distribution can change continuously with increasing population abundance (e.g., Fig. 1e). Although in this case the change in population distribution is gradual, we observe that at intermediate population densities this change can be pretty steep. The steepness depends on the slopes of the payoff functions. An alternative useful way to visualize IFDs is by the use of isodars (Morris 1988). Isodars are curves in patch 1–patch 2 population abundance phase space along which animal fitness is the same in both patches. Morris (2002) used isodars to describe the Allee-type IFD. Figure 3 shows isodars that correspond to IFDs shown in Fig. 1. While isodars for patch payoffs that are negatively density dependent in both patches are continuous lines that intersect with one of the two population axes, the Allee effect introduces a possible gap (called “hiatus” by D. Morris) at low population densities. This is shown in Fig. 3a and corresponds to Figure 2B in Morris (2002). Besides interior distributions there are also two distributions where all of the population occupies one patch only [one of these distributions is missing in Figure 2B in Morris (2002)]. Because of multiple possible IFDs, there are also multiple positive population equilibria. Not all population equilibria (that are also IFDs) are locally asymptotically stable (shown as solid black dots in Fig. 1g–i). Those, where the fitness is positively density dependent are unstable (shown as solid gray dots in Fig. 1g–i). These unstable population equilibria correspond to those IFDs where the patch payoff in one patch is positively density dependent, because in this case the per capita population growth rate is also positively density dependent at such an equilibrium. If the Allee effect is strong, there is the extinction equilibrium which is locally stable. I derived these predictions by assuming that the distribution tracks changes in population numbers instantaneously. I have also shown that these predictions are robust with respect to the relative time scales between population and distributional dynamics when distributional dynamics are described by the replicator equation. This shows that the results are robust with respect to speed of distributional and population processes in this model. In more general models, stability of population equilibria depend on the relation between population and distributional time scales (Cressman and Krivan 2013). The replicator dynamics was recently used by Tran and Cressman (2014) to analyze the IFD under the Allee effect. Their results agree with the static analysis of the habitat selection game given in this article. Acknowledgments I thank Ross Cressman and two anonymous reviewers for their thoughtful suggestions. Institutional support RVO:60077344 is acknowledged.
Appendix A: The IFD First, I study under which conditions strategy u = (1, 0) or (0, 1) is a strict NE at overall population abundance x. These two strategies correspond to population distribution (x11 , x21 ) = (x, 0) and (x12 , x22 ) = (0, x), respectively. A strict NE u ∗ = (u ∗1 , u ∗2 )
123
V. Kˇrivan
satisfies W (u ∗ , u ∗ ; x) > W (u, u ∗ ; x) for any other strategy u = (u 1 , u 2 ) = u ∗ . Thus, u ∗ = (1, 0) is a strict NE provided V1 (x) = min{r1 (1−x/K 1 ), a1 (x−A1 )} > V2 (0) = 1 +r1 ) , min{r2 , −a2 A2 } = −a2 A2 due to assumption (1). For x ≤ x1max = K 1a(a1 K1 A1 +r 1
2 A2 . V1 (x) = a1 (x − A1 ) which implies that u ∗ = (1, 0) is a strict NE when x > a1 A1a−a 1 max ∗ For x > x1 , V1 (x) = r1 (1 − x/K 1 ) which implies that u 1 = 1 is a strict NE 2 A2 2 A2 when x < r1 +a K 1 . Thus, for population densities satisfying a1 A1a−a < x < r1 1
r1 +a2 A2 K 1 , u ∗ = (1, 0) is r1 1 A1 K 2 . It follows x < r2 +a r2 r2 +a1 A1 K2. r2
a strict NE. Similarly, u ∗ = (0, 1) is a strict NE when that u ∗ = (0, 1) is a strict NE when
a2 A2 −a1 A1 a2
< x
0 for all perturbed distributions u = (u 1 , u 2 ) = (u ∗1 , u ∗2 ) in a neighborhood of u ∗ . Let DVi (xi , wi ) denote directional derivative of Vi at xi in a direction wi . Because Vi is piece-wise linear, I get W (u ∗ , u; x) − W (u, u; x) = (u ∗1 − u 1 ) DV1 (x1∗ , x1 − x1∗ ) − DV2 (x2∗ , x2 − x2∗ ) . In particular, when x1∗ = x1max and x2∗ = x2max , DVi (xi∗ , xi − xi∗ ) = Vi (xi∗ )(xi − xi∗ ), and W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1∗ ) + V2 (x2∗ ) x. The distribution (x1∗ , x2∗ ) is an ESS provided the sum of slopes V1 (x1∗ ) + V2 (x2∗ ) is negative. In what follows I study evolutionarily stable distributions as a function of the overall population abundance (for definitions of population thresholds x˜ i see (2)). First, I consider a distribution that satisfies 0 < x1 < x1max and 0 < x2 < x2max . For such a distribution V1 (x1 ) = V2 (x2 ) if a1 (x1 − A1 ) = a2 (x2 − A2 ) which yields (x1 , x2 ) =
a1 A 1 − a2 A 2 + a2 x a2 A 2 − a1 A 1 + a1 x . , a1 + a2 a1 + a2
This distribution exists for population abundances satisfying max
123
a1 A 1 − a2 A 2 a2 A 2 − a1 A 1 , a1 a2
< x < min{x˜ 2 , x˜ 4 }.
The Allee-type ideal free distribution
However, this distribution cannot be an ESS, because the sum of slopes V1 (x1 ) + V2 (x2 ) = a1 + a2 is positive. Second, I consider a distribution that satisfies 0 < x1 < x1max and x2max < x2 . For such a distribution V1 (x1 ) = V2 (x2 ) if a1 (x1 − A1 ) = r2 (1 − x2 /K 2 ) which yields (x13 , x23 ) =
r2 (x − K 2 ) − a1 A1 K 2 K 2 (r2 + a1 (A1 − x)) . , r 2 − a1 K 2 r 2 − a1 K 2
Such a distribution is an ESS provided V1 (x13 ) + V2 (x23 ) = a1 − r2 /K 2 < 0, i.e., a1 < r2 /K 2 . Under this assumption, this distribution satisfies constraints 0 < x13 < x1max and x2max < x23 provided K 2 (a1 A1 + r2 ) < x < min{x˜ 1 , x˜ 2 }. r2 Third, I consider a distribution that satisfies x1max < x1 and x2 < x2max . For such a distribution V1 (x1 ) = V2 (x2 ) if r1 (1 − x1 /K 1 ) = a2 (x2 − A2 ) which yields (x14 , x24 ) =
K 1 (r1 + a2 (A2 − x)) r1 (x − K 1 ) − a2 A2 K 1 , r 1 − a2 K 1 r 1 − a2 K 1
.
This distribution is an ESS provided V1 (x14 ) + V2 (x24 ) = a2 − r1 /K 1 < 0, i.e., a2 < r1 /K 1 . Under this assumption, this distribution satisfies constraints x1max < x14 and x24 < x2max provided K 1 (a2 A2 + r1 ) < x < min{x˜ 3 , x˜ 4 }. r1 Fourth, I consider a distribution that satisfies x1max < x1 and x2max < x2 . For such a distribution V1 (x1 ) = V2 (x2 ) if r1 (1 − x1 /K 1 ) = r2 (1 − x2 /K 2 ) which yields (x15 , x25 ) =
K 1 (K 2 (r1 − r2 ) + r2 x) K 2 (K 1 (r2 − r1 ) + r1 x) . , K 2 r1 + K 1 r2 K 2 r1 + K 1 r2
123
V. Kˇrivan
Because V1 (x15 ) + V2 (x25 ) = −r1 /K 1 − r2 /K 2 < 0, this distribution is always an ESS and satisfies constraints x1max < x15 and x2max < x25 whenever max{x˜ 1 , x˜ 3 } < x. It remains to specify under which conditions population distributions such that either x1∗ = x1max or x2∗ = x2max are IFDs. Let us assume that x1∗ = x1max . Then x2∗ = x − x1max . Provided x > x1max , x2∗ > 0. There are two possibilities: Either x2∗ < x2max or x2∗ > x2max . If x2∗ < x2max then from V1 (x1∗ ) = V2 (x2∗ ) I get that x2∗ = A2 +
r1 a1 (K 1 − A1 ) . a2 (a1 K 1 + r1 )
For x1 < x1max ∗ W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1− ) + V2 (x2∗ ) x = −(u ∗1 − u 1 )2 (a1 + a2 )x < 0, ∗ ) denotes the left derivative. Thus, such a distribution cannot be an ESS. where V1 (x1− Now I consider the situation where x2∗ > x2max . The corresponding distribution is then
x2∗ = K 2 +
a1 r1 K 2 (A1 − K 1 ) . (a1 K 1 + r1 )r2
For x1 < x1max ∗ W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1− ) + V2 (x2∗ ) r2 x = −(u ∗1 − u 1 )2 a1 − K2 and for x1 > x1max ∗ W (u ∗ , u; x) − W (u, u; x) = (u ∗1 − u 1 )2 V1 (x1+ ) + V2 (x2∗ ) r1 r2 ∗ 2 x > 0, = (u 1 − u 1 ) + K1 K2 ∗ ) denotes the right derivative. Thus, distribution (x max , x ∗ ) where x ∗ > where V1 (x1+ 1 2 2 max x2 is an ESS provided a1 < r2 /K 2 . Similar considerations show that distribution (x1∗ , x2max ) is not an ESS when x1∗ < max x1 , while for x1∗ > x1max , the distribution is an ESS provided that a2 < r1 /K 1 . Finally, the distribution (x1max , x2max ) is an ESS provided V1 (x1max ) = V2 (x2max ), a1 < r2 /K 2 and a2 < r1 /K 1 .
123
The Allee-type ideal free distribution
Appendix B: Population equilibrium Now I calculate population equilibria of model (6) assuming population distribution is at the IFD at current population density. (a) For population densities that satisfy (r1 + a2 A2 )K 1 a1 A 1 − a2 A 2 ≤x≤ a1 r1 and the IFD distribution (x11 , x21 ) = (x, 0) population dynamics are dx = a1 x(x − A1 ). dt For x(0) > A1 the population is increasing and for x(0) < A1 the population will die out. These are the typical bi-stable population dynamics caused by the positive dependent population growth. (b) For population densities that satisfy (r2 + a1 A1 )K 2 a2 A 2 − a1 A 1 ≤x≤ a2 r2 and the IFD (x12 , x22 ) = (0, x) population dynamics are dx = a2 x(x − A2 ). dt Qualitatively these are the same population dynamics as in the case (a). (c) For the interior IFD given by (x13 , x23 ) (assuming that a1 < r2 /K 2 ) the corresponding population dynamics (6) are a1 r2 (A1 + K 2 − x) dx =x dt a1 K 2 − r 2 with the interior equilibrium x1∗ = A1 + K 2 . Because a1 < r2 /K 2 , this equilibrium is unstable. (d) For the IFD given by (x14 , x24 ) (assuming a2 < r1 /K 1 ) the corresponding population dynamics (6) are a2 r1 (A2 + K 1 − x) dx =x dt a2 K 1 − r 1 with the interior equilibrium
123
V. Kˇrivan
x1∗ = A2 + K 1 . As I assume a2 < r1 /K 1 , this equilibrium is unstable. (e) For the IFD given by (x15 , x25 ) the corresponding dynamics (6) are r1r2 (K 1 + K 2 − x) dx =x dt K 1 r2 + K 2 r1 with the interior equilibrium x1∗ = K 1 + K 2 . This equilibrium is locally asymptotically stable. Appendix C: Stability of population-distributional equilibria Now I study stability of system (6) and (7). First I consider the Jacobian of this system evaluated at the interior equilibrium (0 < u 1 < 1) where V1 (u 1 x) = V2 (u 2 x) = 0. I will assume that payoffs are differentiable at this point. The Jacobian is u 21 x V1 (u 1 x) + u 22 x V2 (u 2 x), x(u 1 x V1 (u 1 x) − u 2 x V2 (u 2 x)) . u 1 u 2 δ(u 1 V1 (u 1 x) − u 2 V2 (u 2 x)), u 1 u 2 xδ(V1 (u 1 x) + V2 (u 2 x))
J1 =
The trace is Tr J1 = x(u 1 (u 1 + δu 2 )V1 (u 1 x) + u 2 (u 2 + δu 1 )V2 (u 2 x)) and the determinant is det J1 = u 1 u 2 x 2 δV1 (u 1 x)V2 (u 2 x). For local asymptotic stability of the interior equilibrium the trace must be negative and the determinant positive which requires both V1 (u 1 x) and V2 (u 2 x) to be negative. The only locally stable interior equilibrium is x = K 1 + K 2 with the corresponding population distribution u 1 = K 1 /(K 1 + K 2 ). Now I calculate the Jacobian at the boundary equilibrium where u = (1, 0) and x = K1, J2 =
K 1 V1 (K 1 ), K 1 (K 1 V1 (K 1 ) − V2 (0) . 0, δV2 (0)
Since V1 (K 1 ) = −r1 /K 1 < 0, this equilibrium is locally asymptotically stable provided V2 (0) = −a2 A2 < 0. Similar considerations show that the equilibrium where u = (0, 1) and x = K 2 is locally asymptotically stable provided V1 (0) = −a1 A1 < 0.
123
The Allee-type ideal free distribution
Finally, I consider the extinction population equilibrium x = 0. The corresponding Jacobian matrix is 0 u 1 V1 (0) + u 2 V2 (0), . J0 = u 1 u 2 δ(u 1 V1 (0) − u 2 V2 (0)), (u 2 − u 1 )δ(V1 (0) − V2 (0)) The extinction equilibrium is locally asymptotically stable provided u 2 V1 (0) + u 1 V2 (0) < 0 and (u 2 − u 1 )(V1 (0) − V2 (0))(u 1 V1 (0) + u 2 V2 (0)) > 0. It follows that when V2 (0) < V1 (0) < 0, then equilibrium x = 0, u 1 = 1 is locally asymptotically stable. Similarly, if V1 (0) < V2 (0) < 0 then equilibrium x = 0, u 1 = 0 is locally asymptotically stable. References Courchamp F, Berec L, Gascoigne J (2008) Allee effects. Oxford University Press, Oxford Cressman R, Krivan V (2013) Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds. J Math Biol 67:329–358 Cressman R, Kˇrivan V (2006) Migration dynamics for the ideal free distribution. Am Nat 168:384–397 Cressman R, Kˇrivan V (2010) The ideal free distribution as an evolutionarily stable state in densitydependent population games. Oikos 119:1231–1242 Fretwell DS, Lucas HL (1969) On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoret 19:16–32 Fretwell SD (1972) Populations in a seasonal environment. Princeton University Press, Princeton Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge Kˇrivan V, Cressman R (2009) On evolutionary stability in prey-predator models with fast behavioral dynamics. Evol Ecol Res 11:227–251 Kˇrivan V, Cressman R, Schneider S (2008) The ideal free distribution: a review and synthesis of the gametheoretic perspective. Theoret Popul Biol 73:403–425 Morris DW (1988) Habitat-dependent population regulation and community structure. Evol Ecol 2:253–269 Morris DW (2002) Measuring the Allee effect: positive density dependence in small mammals. Ecology 83:14–20 Pohley JH, Thomas B (1983) Non-linear ESS models and frequency dependent selection. Biosystems 16:87–100 Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156 Tran T, Cressman R (2014) The Ideal Free Distribution and Evolutionary Stability in habitat selection games with linear fitness and Allee effect (Mimeo)
123
J. Math. Biol. DOI 10.1007/s00285-013-0742-y
The Allee-type ideal free distribution Vlastimil Kˇrivan
Received: 22 May 2013 / Revised: 20 November 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract The ideal free distribution (IFD) in a two-patch environment where individual fitness is positively density dependent at low population densities is studied. The IFD is defined as an evolutionarily stable strategy of the habitat selection game. It is shown that for low and high population densities only one IFD exists, but for intermediate population densities there are up to three IFDs. Population and distributional dynamics described by the replicator dynamics are studied. It is shown that distributional stability (i.e., IFD) does not imply local stability of a population equilibrium. Thus distributional stability is not sufficient for population stability. Results of this article demonstrate that the Allee effect can strongly influence not only population dynamics, but also population distribution in space. Keywords Dispersal · Evolutionary game theory · Habitat selection game · Logistic equation · Non-smooth analysis · Population dynamics · Optimal foraging Mathematics Subject Classification
26A27 · 92D25 · 92D40 · 92D50
1 Introduction Animal distribution in fragmented landscapes is often described by the ideal free distribution (Fretwell and Lucas 1969; Fretwell 1972). This theory predicts that animals distribute so that payoffs in all occupied patches are the same and maximal. When patch payoffs are negatively density dependent the IFD is an evolutionarily stable strategy of the habitat selection game (Kˇrivan et al. 2008). The habitat selection game
V. Kˇrivan (B) Biology Center, Academy of Sciences of the Czech Republic, and Faculty of Science, University of South Bohemia, Branišovská 31, 370 05 Ceske Budejovice, Czech Republic e-mail: [email protected]
123
V. Kˇrivan
is a game theoretical concept (similar to the Hawk–Dove game or to the Prisoner’s dilemma) that seeks evolutionarily stable animal distributions in space. In their pioneering article Fretwell and Lucas (1969) also briefly mentioned the Allee-type IFD where patch payoffs are positively density dependent at low population densities. In particular, using graphical analysis they predicted that a small change in population abundance can lead to an abrupt (discontinuous) change in population distribution. For example, when all individuals occupy patch 1 at low population densities there can be a population threshold at which population distribution changes discontinuously. Morris (2002) used the Allee-type IFD to study distributions of the deer mice. While the Fretwell and Lucas’ geometric analysis assumed a general hump-shaped patch payoff function, Morris (2002) considered payoffs that first linearly increase and then linearly decrease with population abundance. Such piece-wise linear payoff functions qualitatively capture properties of many Allee-type fitness functions used in the literature (Courchamp et al. 2008), and they are easier to analyze when compared with non-linear functions. Both Fretwell and Lucas (1969) and Morris (2002) assumed a particular case where the payoff in one patch is always higher than the payoff in the other patch. In other words, the graphs of these two payoffs never intersected (e.g., see Figure 2 in Morris 2002). Fretwell and Lucas (1969) concluded by “One can imagine [payoff] curves which lead to complete shifts in population [distribution] while other curves may lead to no erratic behavior at all”. In this article I will analyze such more general patch payoffs and will provide conditions under which erratic changes in distributions do occur. I will show that for some intermediate population densities there are up to three possible IFDs and there are three alternative population equilibria. I will also study population and distributional dynamics under the assumption that individuals disperse in the direction of increasing fitness. 2 Model I consider a single population in a heterogeneous environment consisting of two patches. Following Morris (2002) I assume piecewise linear patch payoffs (Fig. 1a) xi , i = 1, 2, Vi (xi ) = min ai (xi − Ai ) , ri 1 − Ki where xi is the population abundance in patch i, ri is the intrinsic per capita population growth rate, K i is the patch carrying capacity, Ai is the Allee threshold, and ai is the strength of positive density dependence at low population densities. I will assume that, except the Allee threshold, all parameters are positive. The Allee threshold is the population density below which the population growth rate is negative and above which it is positive. If the Allee threshold is positive, the Allee effect is strong, because at low population densities the per capita population growth rate is negative and the population dies out. If the Allee threshold is negative, then the Allee effect is weak because even at low population densities the per capita population growth rate is positive. Function Vi has one maximum at population density ximax =
123
K i (ai Ai + ri ) , i = 1, 2 ai K i + ri
The Allee-type ideal free distribution
at which the patch payoff is Vi (ximax ) =
ai ri (K i − Ai ) , i = 1, 2. ai K i + ri
I will assume throughout the article that ximax > 0 (i = 1, 2) which means ai Ai + ri > 0,
i = 1, 2.
(1)
If the opposite inequality holds, patch payoffs would be negatively density dependent for all population densities (i.e., no Allee effect). In addition, I will assume that Vi (ximax ) > 0 (i.e., K i > Ai ) so that the payoff functions have a triangular form (Fig. 1a–c) qualitatively similar to other models of the per capita population growth rate with the Allee effect used in the literature (see Table 3.1 on p. 68 in Courchamp et al. 2008). In what follows I am interested in the distribution (x1 , x2 ) of total x = x1 + x2 individuals among the two habitats. If u i denotes the proportion of the lifetime an average individual spends in patch i = 1, 2 and the population is monomorphic then xi = u i x. Thus, the proportion of time spend in each of the two patches (u 1 , u 2 ) (u 1 + u 2 = 1) is the individual strategy. Fitness of a mutant with strategy u˜ = (u˜ 1 , u˜ 2 ) in the resident population with strategy u = (u 1 , u 2 ) is calculated as the average payoff W (u, ˜ u; x) = u˜ 1 V1 (u 1 x) + u˜ 2 V2 (u 2 x). The underlying game was called the habitat selection game (HSG; Kˇrivan et al. 2008; Kˇrivan and Cressman 2009; Cressman and Kˇrivan 2010). The Nash equilibria of the HSG coincide with the ideal free distribution concept [IFD; Fretwell and Lucas (1969); Kˇrivan et al. (2008)]. When patch payoffs are negatively density dependent, there is a unique Nash equilibrium of the HSG which is also an evolutionarily stable strategy (Cressman and Kˇrivan 2006; Kˇrivan et al. 2008). In this article I will study evolutionarily stable strategies (called here IFDs) (u ∗1 , u ∗2 ) for fitness function W. Because the fitness function is non-linear in the resident strategy u = (u 1 , u 2 ), I use the local evolutionary stability condition (Pohley and Thomas 1983; Hofbauer and Sigmund 1998) that classifies strategy u ∗ = (u ∗1 , u ∗2 ) as an (local) ESS if W (u ∗ , u; x) > W (u, u; x) for all strategies u = (u 1 , u 2 ) = (u ∗1 , u ∗2 ) = u ∗ in a neighborhood of u ∗ . On the other hand, since W (u ∗ , u; x) is linear in u ∗1 and u ∗2 , patch payoffs must be equal at an ESS (i.e. V1 (u ∗1 x) = V2 (u ∗2 x)) if 0 < u ∗1 < 1 (Hofbauer and Sigmund 1998). In what follows I will need the following threshold total patch-wide population abundances: x˜ 1 =
a1 K 2 r1 (A1 − K 1 ) + a1 K 1r2 (A1 + K 2 ) + r1r2 (K 1 + K 2 ) , r2 (a1 K 1 + r1 )
123
123
30
40
50
0
10
20
30
0
10
20
30 40 50
H
10
0
10
20
0
0
30
0.0
0.0
20
0.2
1.0
0.2
E
0.4
40
0.4
30
0.6
20
0.6
10
10
0.5
0.0
0.5
1.0
1.5
2.0
0.8
0
0
B
0.8
1.0
2
1
0
1
2
0
10
10
5
20
20
10
30
30
15
40
40
20
50
50
25
I
F
C
0
50
0
50
0
0
100
0.0
0.2
0.4
0.6
0.8
1.0
1
0
1
2
10
10
5
20
20
10
30
30
15
40
40
20
50
50
25
60
60
30
70
70
35
Fig. 1 a–c Payoffs V1 (solid line) and V2 (dashed line) as a function of x1 and x2 , respectively. d–f Show corresponding IFDs (u 1 ; here segments u i1 = x1i /x, i = 1, . . . , 5 correspond to distributions (a)–(e) defined in the text) as a function of total population abundance x. g–i The population growth rate x1 V1 (x1 ) + x2 V2 (x2 ) under the IFD. The black dots denote locally stable population equilibria, while light gray dots are unstable population equilibria. The line styles in g–i corresponds to those in d–f. Parameters used in a, d and g: r1 = 3.5, r2 = 2.5, a1 = 0.25, a2 = 0.3, K 1 = 20, K 2 = 25, A1 = 1.5, A2 = 3.5. Parameters used in b, e and h: r1 = 3.5, r2 = 2., K 1 = 20, K 2 = 25, A1 = 1.5, A2 = 3.5, a1 = 0.25, a2 = 0.15, Parameters used in c, f and i: r1 = 5, r2 = 10, K 1 = 35, K 2 = 30, A1 = 9, A2 = 2, a1 = 0.15, a2 = 0.1
G
D
A
V. Kˇrivan
The Allee-type ideal free distribution
a2 r2 (K 2 − A2 ) + a1r2 (A1 + K 2 ) + a1 a2 K 2 (A1 + A2 ) , a1 (a2 K 2 + r2 ) a2 K 2 r1 (A2 + K 1 ) + a2 r2 K 1 (A2 − K 2 ) + r1r2 (K 1 + K 2 ) x˜ 3 = , r1 (a2 K 2 + r2 ) a1 a2 K 1 (A1 + A2 ) + a1 r1 (K 1 − A1 ) + a2 r1 (A2 + K 1 ) . x˜ 4 = a2 (a1 K 1 + r1 )
x˜ 2 =
(2)
These abundances correspond to population distributions at which patch payoffs are the same (i.e., V1 (x1 ) = V2 (x2 )) and the patch payoff at least in one patch is maximal (i.e., the population density in at least one patch equals ximax ). In particular x˜ 1 assumes that V2 (x2max ) exceeds V1 (x1max ). In this case, the equation V1 (x1max ) = V2 (x2 ) always has (A1 −K 1 )K 2 r1 solution x2 = K 2 + a1(a which is in the range of population densities at which 1 K 1 +r1 )r2 the patch 2 payoff is negatively density dependent. For example, in Fig. 1c, x1max = 21.7 and x2 = 24.3 so that x˜ 1 = 46.0. When V2 (0) < V1 (x1max ), equation V1 (x1max ) = 1 −A1 ) V2 (x2 ) has another solution x2 = A2 + aa12r(a1 (K (e.g., x2 = 21.0 in Fig. 1c) 1 K 1 +r1 ) which is in the range of population densities where the patch 2 payoff is positively density dependent. This population distribution (x1max , x2 ) defines the total population threshold x˜ 4 . Similarly, x˜ 3 assumes that V1 (x1max ) exceeds V2 (x2max ) (Fig. 1a, b). In this 2 r2 (A2 −K 2 ) case, the equation V1 (x1 ) = V2 (x2max ) always has solution x1 = K 1 + K r11a(a 2 K 2 +r2 ) which is in the range of population densities at which the patch 1 payoff is negatively density dependent. When V1 (0) < V2 (x2max ), equation V1 (x1 ) = V2 (x2max ) has another 2 −A2 ) solution x1 = A1 + aa21r(a2 (K which is in the range of population densities at 2 K 2 +r2 ) which the patch 1 payoff is positively density dependent. This population distribution (x1 , x2max ) defines the total population threshold x˜ 2 . For example, in Fig. 1b, x1 = 5.9 and x2max = 10.9 so that x˜ 2 = 16.8. Appendix A shows that there are the following IFDs: (a) For population densities that satisfy (r1 + a2 A2 )K 1 a1 A 1 − a2 A 2 ≤x≤ a1 r1 distribution (x11 , x21 ) = (x, 0) where all individuals occupy patch 1 is an IFD. E.g., when patch 1 is more profitable than patch 2 when both patches are unoccupied, i.e., V1 (0) = −a1 A1 > V2 (0) = −a2 A2 , (1) implies that r1 + a2 A2 > 0 and the above inequality holds for small population densities. (b) For population densities that satisfy (r2 + a1 A1 )K 2 a2 A 2 − a1 A 1 ≤x≤ a2 r2 distribution (x12 , x22 ) = (0, x) is an IFD and all individuals occupy patch 2 only. (c) For population densities that satisfy K 2 (a1 A1 + r2 ) < x < min{x˜ 1 , x˜ 2 }, r2
123
V. Kˇrivan
distribution (x13 , x23 )
=
r2 (x − K 2 ) − a1 A1 K 2 K 2 (r2 + a1 (A1 − x)) , r 2 − a1 K 2 r 2 − a1 K 2
(3)
is an IFD provided a1 < r2 /K 2 . At this distribution density dependence in patch 1 is positive and density dependence in patch 2 is negative (i.e., 0 < x13 ≤ x1max and x2max ≤ x23 < x). (d) For population densities that satisfy K 1 (a2 A2 + r1 ) < x < min{x˜ 3 , x˜ 4 }, r1 distribution (x14 , x24 )
=
K 1 (r1 + a2 (A2 − x)) r1 (x − K 1 ) − a2 A2 K 1 , r 1 − a2 K 1 r 1 − a2 K 1
(4)
is an IFD provided a2 < r1 /K 1 . At this distribution density dependence in patch 1 is negative and density dependence in patch 2 is positive (i.e., x1max ≤ x14 and 0 < x24 ≤ x2max ). (e) For population densities that satisfy max{x˜ 1 , x˜ 3 } < x, distribution (x15 , x25 ) =
K 1 (K 2 (r1 − r2 ) + r2 x) K 2 (K 1 (r2 − r1 ) + r1 x) , K 2 r1 + K 1 r2 K 2 r1 + K 1 r2
(5)
is an IFD. At this equilibrium density dependence is negative at both patches (i.e., x1max < x15 and x2max < x25 ). A. It The case where either x1 = x1max or x2 = x2max is discussed in Appendix a1 r1 K 2 (A1 −K 1 ) max is an IFD is shown there that population distribution x1 , K 2 + (a1 K 1 +r1 )r2 −K 2 ) max is an if a1 < r2 /K 2 . Similarly, population distribution K 1 + a2rr12(aK21 K(A22+r , x 2 2) IFD provided a2 < r1 /K 1 . Population distribution (x1max , x2max ) is an IFD provided a1 < r2 /K 2 and a2 < r1 /K 1 . Dependence of these IFDs on population abundance is shown in Fig. 1d–f, where u i1 = x1i /x, i = 1, . . . , 5. For low population densities all individuals occupy the patch that has the higher payoff when unoccupied. It is patch 1 in Fig. 1d, e, where V1 (0) > V2 (0) and patch 2 in Fig. 1f where V1 (0) < V2 (0). Because at low population densities patch payoffs are positively density dependent, individuals prefer to stay together in the more profitable patch as population abundance increases. Due to initial positive density dependence of patch payoffs there is also an alternative distribution at slightly higher population abundances under which all individuals occupy the patch
123
The Allee-type ideal free distribution
that was initially inferior (this is patch 2 in Fig. 1d, e and patch 1 in Fig. 1f). The minimal population density for which this alternative distribution is possible satisfies V2 (x) = V1 (0) in Fig. 1d and e, and V1 (x) = V2 (0) in Fig. 1f. This is because when the originally poorer patch is occupied by a population that is abundant enough, its patch payoff will be higher than in the unoccupied patch. As the overall population density increases, the payoff in the occupied patch reaches its maximum and starts to decline due to negative density dependence. When the decreasing payoff in the occupied patch reaches the value that equals the payoff in the unoccupied patch there are two qualitatively different possibilities. Let us consider the case where all individuals occupy patch 1 in Fig. 1d (the solid line). At the critical population density x = 26, V1 (x) = V2 (0) (Fig. 1a). I remark, that in this case population distributions (x13 , x23 ) and (x14 , x24 ) do not exist because K 2 (a1 A1 + r2 )/r2 = 28.75 is greater than is min{x˜ 1 , x˜ 2 } = min{15.07, 16.82} and K 1 (a2 A2 + r1 )/r1 = 26 is greater than min{x˜ 3 , x˜ 4 } = min{19.66, 18.96}. Thus, the only possible distribution for x ≥ 26 is the interior distribution (x15 , x25 ) (the short dashed line denoted by u 51 in Fig. 1d). Because for x = 26 this distribution is (x15 , x25 ) = (13, 13) there is a discontinuity in Fig. 1d at x = 26. Now I will consider the situation shown in Fig. 1e where at the critical population density x = 23 distribution at which all individuals occupy patch 1 only changes continuously to a distribution where both patches are occupied. This is because in this plot parameters are such that K 1 (a2 A2 + r1 )/r1 = 23 < min{x˜ 3 , x˜ 4 } = min{24.5, 25.3} and a2 = 0.15 < r1 /K 1 = 0.175. Thus, as population density increases above 23, the population will continuously re-distribute over both patches following distribution (x14 , x24 ). Under this distribution the payoffs in both patches will increase until population density in patch 2 will reach x2max . This happens when the overall population density equals x˜ 3 = 24.5. From then on, payoffs in both patches will decrease and the distribution will continuously change according to distribution (x15 , x25 ). The case shown in Fig. 1f where all individuals occupy patch 1 only is similar, but here distribution (x14 , x24 ) changes discontinuously to (x13 , x23 ). This is because payoffs in both patches increase for increasing population densities until the population density in patch 1 reaches x1max . At this moment the corresponding distribution in the second patch is still in the region where patch 2 is positively density dependent. Consequently, this distribution cannot be continuously extended for higher population densities. 3 Population dynamics The overall population dynamics are described by the following differential equation dx = u 1 x V1 (u 1 x) + u 2 x V2 (u 2 x) = x V dt
(6)
where (u 1 , u 2 ) denotes the population distribution among the two patches. This population model is an extension of a population growth with the Allee effect to two-patch environments. When u 1 and u 2 are fixed, i.e., when individual strategy is non-adaptive, the population growth rate is shown in Fig. 2. Provided the Allee effect is strong
123
V. Kˇrivan
30
20
10
0 0
10
20
30
40
50
Fig. 2 Population growth rate (x1 V1 (x1 ) + x2 V2 (x2 )) in model (6) for a fixed population distribution (u 1 , u 2 ) = (0.6, 0.4). Other parameters are the same as those used in Fig. 1a
(u 1 V1 (0) + u 2 V2 (0) = −u 1 a1 A1 − u 2 a2 A2 < 0), for low initial population densities population dynamics converge to extinction, while above the Allee threshold population dynamics tend to a positive equilibrium. Here I consider the situation where the population distributes very fast according to the IFD derived in the previous section. In other words, dispersal dynamics are assumed to be (infinitely) fast when compared with population dynamics. From the previous section it follows that there are up to three IFDs at a given positive population density. Figure 1g–i shows the mean population growth rate (x V ) for each possible IFD (shown in Fig. 1d–f; the line type correspond in these panels). Only those equilibria where the population growth rate is negatively density dependent are locally asymptotically stable (black dots). Thus, there are up to three locally stable positive population equilibria. If all individuals are in patch 1 (patch 2) only, the corresponding positive equilibrium is K 1 (K 2 ). When the population occupies both habitats, the only positive population equilibrium is K 1 + K 2 which corresponds to the IFD where both patches are occupied up to their carrying capacities. It is interesting to note that population equilibria that correspond to IFDs at which the payoff in one patch increases and decreases in the other patch (these IFDs are given by (3) or (4) and correspond to the two middle length dashed curves in Fig. 1e, f) can never be stable, because the per capita population growth rate at these distributions is always positively density dependent (see Fig. 1h, i where corresponding unstable equilibria are x ∼ 23 in Fig. 1h and x ∼ 37 and x ∼ 39 in Fig. 1i). This shows that distributional stability is different from population stability. In other words, population dynamics can destabilize distributions that are evolutionarily stable at a fixed population density. When population and distributional time scales cannot be separated, i.e., when both processes operate on similar time-scales, distributional dynamics can be described by the replicator equation (Taylor and Jonker 1978; Hofbauer and Sigmund 1998; Tran and Cressman 2014) du 1 = δu 1 (1 − u 1 )(V1 (u 1 x) − V2 (u 2 x)) dt
123
(7)
The Allee-type ideal free distribution
where u 2 = 1 − u 1 , and δ > 0 shows how demographic and distributional time scales are related (Cressman and Krivan 2013). This equation together with population dynamics (6) describes linked changes in population dynamics and distribution. It is proved in Appendix C that the extinction equilibrium (x = 0) is locally asymptotically stable when V1 (0) + V2 (0) = −a1 A1 − a2 A2 < 0. When V1 (0) = −a1 A1 < 0 (V2 (0) = −a2 A2 < 0) then the equilibrium at which all individuals occupy patch 2 (patch 1) only is locally asymptotically stable. Finally, the interior equilibrium where x1 = K 1 , x2 = K 2 , u 1 = K 1 /(K 1 + K 2 ), and u 2 = K 2 /(K 1 + K 2 ) is always locally asymptotically stable. These predictions do not differ from those where I assume that the population distribution is at the IFD at the current population density, i.e., when distributional dynamics are infinitely faster when compared with population dynamics. However, for more complex models population stability can depend on time scales (Cressman and Krivan 2013).
4 Discussion In this article I have studied the Allee-type ideal free distribution. I have shown that depending on parameters and population densities the Allee effect leads to multiple IFDs. They correspond to distributions where all individuals occupy patch 1 only, patch 2 only, or both patches. When population dynamics are considered, there can be up to three locally stable population equilibria. I have also shown that evolutionary stability of a population distribution does not guarantee population stability, because a population equilibrium which is also an IFD does not need to be locally stable. The IFD was introduced by Fretwell and Lucas (1969) who defined it as a Nash equilibrium. When density dependence is negative, it was shown (Cressman and Kˇrivan 2006; Kˇrivan et al. 2008) that there is a unique IFD that is also an evolutionarily stable strategy of the underlying habitat selection game. This means that this unique IFD is resistant to small perturbations. Fretwell and Lucas (1969) also considered the Allee-type IFD where density dependence is positive at low population densities. They showed that the IFD can dramatically (discontinuously) change at some critical overall population densities. They concluded that “...species following Allee’s principle may demonstrate erratic changes in distribution with small changes in population.” Their work was continued by Morris (2002) who analyzed such erratic changes in distribution of the deer mice. Both these works assumed patch payoffs without an intersection and the analysis was purely graphical. While Fretwell and Lucas (1969) assumed smooth patch payoffs, Morris (2002) assumed piece-wise linear patch payoffs. Such piecewise linear patch payoffs qualitatively capture properties of many hump shaped fitness functions used in the literature on the Allee effect (Courchamp et al. 2008). The advantage of piece-wise linear functions when compared with non-linear functions used in the literature on the Allee effect is that they are more readily analyzable. In this article, following Morris (2002), I have analyzed the general case where patch payoffs are piece-wise linear and they can intersect. Depending on model parameters and overall population density I have analyzed all corresponding IFDs. I have shown that up to three IFDs can co-exist at a single population abundance. However, this can happen only for intermediate population densities where the Allee effect influences
123
V. Kˇrivan Fig. 3 Isodars corresponding to IFDs shown in Fig. 1d–f. Isodars connect population numbers in patch 1 and patch 2 phase space along which the expected fitness is the same in both patches (i.e., V1 (x1 ) = V2 (x2 )). The lines along the coordinate axes correspond to distributions where the population occupies one patch only. When population numbers are low, the interior isodar does not exists (a) because all individuals occupy patch 1 or patch 2 only due to the Allee effect. Depending on patch payoffs, the interior isodar can be piece-wise linear (b, c). The line types in these panels correspond to line types in Fig. 1d–f. Parameters are the same as those used in Fig. 1
A 30 25 20 15 10 5 0 0
5
10
15
20
25
30
0
5
10
15
20
25
30
B 40
30
20
10
0
C 40
30
20
10
0 0
10
20
30
40
the population distribution at least in one patch. At high population densities where both patch payoffs are negatively density dependent there is a unique IFD (Fig. 1d–f). At intermediate population densities two pure strategies where all individuals occupy patch 1 or patch 2 only exist. At very low population densities only the patch with the highest payoff when unoccupied is populated. Whether there will be discontinuous
123
The Allee-type ideal free distribution
(erratic) changes in the IFD as overall population density increases depends on the history and slopes of the two payoff functions. For example, in Fig. 1a and d, as population size increases, there will be a sudden change at the IFD. However, this is not always so because the population distribution can change continuously with increasing population abundance (e.g., Fig. 1e). Although in this case the change in population distribution is gradual, we observe that at intermediate population densities this change can be pretty steep. The steepness depends on the slopes of the payoff functions. An alternative useful way to visualize IFDs is by the use of isodars (Morris 1988). Isodars are curves in patch 1–patch 2 population abundance phase space along which animal fitness is the same in both patches. Morris (2002) used isodars to describe the Allee-type IFD. Figure 3 shows isodars that correspond to IFDs shown in Fig. 1. While isodars for patch payoffs that are negatively density dependent in both patches are continuous lines that intersect with one of the two population axes, the Allee effect introduces a possible gap (called “hiatus” by D. Morris) at low population densities. This is shown in Fig. 3a and corresponds to Figure 2B in Morris (2002). Besides interior distributions there are also two distributions where all of the population occupies one patch only [one of these distributions is missing in Figure 2B in Morris (2002)]. Because of multiple possible IFDs, there are also multiple positive population equilibria. Not all population equilibria (that are also IFDs) are locally asymptotically stable (shown as solid black dots in Fig. 1g–i). Those, where the fitness is positively density dependent are unstable (shown as solid gray dots in Fig. 1g–i). These unstable population equilibria correspond to those IFDs where the patch payoff in one patch is positively density dependent, because in this case the per capita population growth rate is also positively density dependent at such an equilibrium. If the Allee effect is strong, there is the extinction equilibrium which is locally stable. I derived these predictions by assuming that the distribution tracks changes in population numbers instantaneously. I have also shown that these predictions are robust with respect to the relative time scales between population and distributional dynamics when distributional dynamics are described by the replicator equation. This shows that the results are robust with respect to speed of distributional and population processes in this model. In more general models, stability of population equilibria depend on the relation between population and distributional time scales (Cressman and Krivan 2013). The replicator dynamics was recently used by Tran and Cressman (2014) to analyze the IFD under the Allee effect. Their results agree with the static analysis of the habitat selection game given in this article. Acknowledgments I thank Ross Cressman and two anonymous reviewers for their thoughtful suggestions. Institutional support RVO:60077344 is acknowledged.
Appendix A: The IFD First, I study under which conditions strategy u = (1, 0) or (0, 1) is a strict NE at overall population abundance x. These two strategies correspond to population distribution (x11 , x21 ) = (x, 0) and (x12 , x22 ) = (0, x), respectively. A strict NE u ∗ = (u ∗1 , u ∗2 )
123
V. Kˇrivan
satisfies W (u ∗ , u ∗ ; x) > W (u, u ∗ ; x) for any other strategy u = (u 1 , u 2 ) = u ∗ . Thus, u ∗ = (1, 0) is a strict NE provided V1 (x) = min{r1 (1−x/K 1 ), a1 (x−A1 )} > V2 (0) = 1 +r1 ) , min{r2 , −a2 A2 } = −a2 A2 due to assumption (1). For x ≤ x1max = K 1a(a1 K1 A1 +r 1
2 A2 . V1 (x) = a1 (x − A1 ) which implies that u ∗ = (1, 0) is a strict NE when x > a1 A1a−a 1 max ∗ For x > x1 , V1 (x) = r1 (1 − x/K 1 ) which implies that u 1 = 1 is a strict NE 2 A2 2 A2 when x < r1 +a K 1 . Thus, for population densities satisfying a1 A1a−a < x < r1 1
r1 +a2 A2 K 1 , u ∗ = (1, 0) is r1 1 A1 K 2 . It follows x < r2 +a r2 r2 +a1 A1 K2. r2
a strict NE. Similarly, u ∗ = (0, 1) is a strict NE when that u ∗ = (0, 1) is a strict NE when
a2 A2 −a1 A1 a2
< x
0 for all perturbed distributions u = (u 1 , u 2 ) = (u ∗1 , u ∗2 ) in a neighborhood of u ∗ . Let DVi (xi , wi ) denote directional derivative of Vi at xi in a direction wi . Because Vi is piece-wise linear, I get W (u ∗ , u; x) − W (u, u; x) = (u ∗1 − u 1 ) DV1 (x1∗ , x1 − x1∗ ) − DV2 (x2∗ , x2 − x2∗ ) . In particular, when x1∗ = x1max and x2∗ = x2max , DVi (xi∗ , xi − xi∗ ) = Vi (xi∗ )(xi − xi∗ ), and W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1∗ ) + V2 (x2∗ ) x. The distribution (x1∗ , x2∗ ) is an ESS provided the sum of slopes V1 (x1∗ ) + V2 (x2∗ ) is negative. In what follows I study evolutionarily stable distributions as a function of the overall population abundance (for definitions of population thresholds x˜ i see (2)). First, I consider a distribution that satisfies 0 < x1 < x1max and 0 < x2 < x2max . For such a distribution V1 (x1 ) = V2 (x2 ) if a1 (x1 − A1 ) = a2 (x2 − A2 ) which yields (x1 , x2 ) =
a1 A 1 − a2 A 2 + a2 x a2 A 2 − a1 A 1 + a1 x . , a1 + a2 a1 + a2
This distribution exists for population abundances satisfying max
123
a1 A 1 − a2 A 2 a2 A 2 − a1 A 1 , a1 a2
< x < min{x˜ 2 , x˜ 4 }.
The Allee-type ideal free distribution
However, this distribution cannot be an ESS, because the sum of slopes V1 (x1 ) + V2 (x2 ) = a1 + a2 is positive. Second, I consider a distribution that satisfies 0 < x1 < x1max and x2max < x2 . For such a distribution V1 (x1 ) = V2 (x2 ) if a1 (x1 − A1 ) = r2 (1 − x2 /K 2 ) which yields (x13 , x23 ) =
r2 (x − K 2 ) − a1 A1 K 2 K 2 (r2 + a1 (A1 − x)) . , r 2 − a1 K 2 r 2 − a1 K 2
Such a distribution is an ESS provided V1 (x13 ) + V2 (x23 ) = a1 − r2 /K 2 < 0, i.e., a1 < r2 /K 2 . Under this assumption, this distribution satisfies constraints 0 < x13 < x1max and x2max < x23 provided K 2 (a1 A1 + r2 ) < x < min{x˜ 1 , x˜ 2 }. r2 Third, I consider a distribution that satisfies x1max < x1 and x2 < x2max . For such a distribution V1 (x1 ) = V2 (x2 ) if r1 (1 − x1 /K 1 ) = a2 (x2 − A2 ) which yields (x14 , x24 ) =
K 1 (r1 + a2 (A2 − x)) r1 (x − K 1 ) − a2 A2 K 1 , r 1 − a2 K 1 r 1 − a2 K 1
.
This distribution is an ESS provided V1 (x14 ) + V2 (x24 ) = a2 − r1 /K 1 < 0, i.e., a2 < r1 /K 1 . Under this assumption, this distribution satisfies constraints x1max < x14 and x24 < x2max provided K 1 (a2 A2 + r1 ) < x < min{x˜ 3 , x˜ 4 }. r1 Fourth, I consider a distribution that satisfies x1max < x1 and x2max < x2 . For such a distribution V1 (x1 ) = V2 (x2 ) if r1 (1 − x1 /K 1 ) = r2 (1 − x2 /K 2 ) which yields (x15 , x25 ) =
K 1 (K 2 (r1 − r2 ) + r2 x) K 2 (K 1 (r2 − r1 ) + r1 x) . , K 2 r1 + K 1 r2 K 2 r1 + K 1 r2
123
V. Kˇrivan
Because V1 (x15 ) + V2 (x25 ) = −r1 /K 1 − r2 /K 2 < 0, this distribution is always an ESS and satisfies constraints x1max < x15 and x2max < x25 whenever max{x˜ 1 , x˜ 3 } < x. It remains to specify under which conditions population distributions such that either x1∗ = x1max or x2∗ = x2max are IFDs. Let us assume that x1∗ = x1max . Then x2∗ = x − x1max . Provided x > x1max , x2∗ > 0. There are two possibilities: Either x2∗ < x2max or x2∗ > x2max . If x2∗ < x2max then from V1 (x1∗ ) = V2 (x2∗ ) I get that x2∗ = A2 +
r1 a1 (K 1 − A1 ) . a2 (a1 K 1 + r1 )
For x1 < x1max ∗ W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1− ) + V2 (x2∗ ) x = −(u ∗1 − u 1 )2 (a1 + a2 )x < 0, ∗ ) denotes the left derivative. Thus, such a distribution cannot be an ESS. where V1 (x1− Now I consider the situation where x2∗ > x2max . The corresponding distribution is then
x2∗ = K 2 +
a1 r1 K 2 (A1 − K 1 ) . (a1 K 1 + r1 )r2
For x1 < x1max ∗ W (u ∗ , u; x) − W (u, u; x) = −(u ∗1 − u 1 )2 V1 (x1− ) + V2 (x2∗ ) r2 x = −(u ∗1 − u 1 )2 a1 − K2 and for x1 > x1max ∗ W (u ∗ , u; x) − W (u, u; x) = (u ∗1 − u 1 )2 V1 (x1+ ) + V2 (x2∗ ) r1 r2 ∗ 2 x > 0, = (u 1 − u 1 ) + K1 K2 ∗ ) denotes the right derivative. Thus, distribution (x max , x ∗ ) where x ∗ > where V1 (x1+ 1 2 2 max x2 is an ESS provided a1 < r2 /K 2 . Similar considerations show that distribution (x1∗ , x2max ) is not an ESS when x1∗ < max x1 , while for x1∗ > x1max , the distribution is an ESS provided that a2 < r1 /K 1 . Finally, the distribution (x1max , x2max ) is an ESS provided V1 (x1max ) = V2 (x2max ), a1 < r2 /K 2 and a2 < r1 /K 1 .
123
The Allee-type ideal free distribution
Appendix B: Population equilibrium Now I calculate population equilibria of model (6) assuming population distribution is at the IFD at current population density. (a) For population densities that satisfy (r1 + a2 A2 )K 1 a1 A 1 − a2 A 2 ≤x≤ a1 r1 and the IFD distribution (x11 , x21 ) = (x, 0) population dynamics are dx = a1 x(x − A1 ). dt For x(0) > A1 the population is increasing and for x(0) < A1 the population will die out. These are the typical bi-stable population dynamics caused by the positive dependent population growth. (b) For population densities that satisfy (r2 + a1 A1 )K 2 a2 A 2 − a1 A 1 ≤x≤ a2 r2 and the IFD (x12 , x22 ) = (0, x) population dynamics are dx = a2 x(x − A2 ). dt Qualitatively these are the same population dynamics as in the case (a). (c) For the interior IFD given by (x13 , x23 ) (assuming that a1 < r2 /K 2 ) the corresponding population dynamics (6) are a1 r2 (A1 + K 2 − x) dx =x dt a1 K 2 − r 2 with the interior equilibrium x1∗ = A1 + K 2 . Because a1 < r2 /K 2 , this equilibrium is unstable. (d) For the IFD given by (x14 , x24 ) (assuming a2 < r1 /K 1 ) the corresponding population dynamics (6) are a2 r1 (A2 + K 1 − x) dx =x dt a2 K 1 − r 1 with the interior equilibrium
123
V. Kˇrivan
x1∗ = A2 + K 1 . As I assume a2 < r1 /K 1 , this equilibrium is unstable. (e) For the IFD given by (x15 , x25 ) the corresponding dynamics (6) are r1r2 (K 1 + K 2 − x) dx =x dt K 1 r2 + K 2 r1 with the interior equilibrium x1∗ = K 1 + K 2 . This equilibrium is locally asymptotically stable. Appendix C: Stability of population-distributional equilibria Now I study stability of system (6) and (7). First I consider the Jacobian of this system evaluated at the interior equilibrium (0 < u 1 < 1) where V1 (u 1 x) = V2 (u 2 x) = 0. I will assume that payoffs are differentiable at this point. The Jacobian is u 21 x V1 (u 1 x) + u 22 x V2 (u 2 x), x(u 1 x V1 (u 1 x) − u 2 x V2 (u 2 x)) . u 1 u 2 δ(u 1 V1 (u 1 x) − u 2 V2 (u 2 x)), u 1 u 2 xδ(V1 (u 1 x) + V2 (u 2 x))
J1 =
The trace is Tr J1 = x(u 1 (u 1 + δu 2 )V1 (u 1 x) + u 2 (u 2 + δu 1 )V2 (u 2 x)) and the determinant is det J1 = u 1 u 2 x 2 δV1 (u 1 x)V2 (u 2 x). For local asymptotic stability of the interior equilibrium the trace must be negative and the determinant positive which requires both V1 (u 1 x) and V2 (u 2 x) to be negative. The only locally stable interior equilibrium is x = K 1 + K 2 with the corresponding population distribution u 1 = K 1 /(K 1 + K 2 ). Now I calculate the Jacobian at the boundary equilibrium where u = (1, 0) and x = K1, J2 =
K 1 V1 (K 1 ), K 1 (K 1 V1 (K 1 ) − V2 (0) . 0, δV2 (0)
Since V1 (K 1 ) = −r1 /K 1 < 0, this equilibrium is locally asymptotically stable provided V2 (0) = −a2 A2 < 0. Similar considerations show that the equilibrium where u = (0, 1) and x = K 2 is locally asymptotically stable provided V1 (0) = −a1 A1 < 0.
123
The Allee-type ideal free distribution
Finally, I consider the extinction population equilibrium x = 0. The corresponding Jacobian matrix is 0 u 1 V1 (0) + u 2 V2 (0), . J0 = u 1 u 2 δ(u 1 V1 (0) − u 2 V2 (0)), (u 2 − u 1 )δ(V1 (0) − V2 (0)) The extinction equilibrium is locally asymptotically stable provided u 2 V1 (0) + u 1 V2 (0) < 0 and (u 2 − u 1 )(V1 (0) − V2 (0))(u 1 V1 (0) + u 2 V2 (0)) > 0. It follows that when V2 (0) < V1 (0) < 0, then equilibrium x = 0, u 1 = 1 is locally asymptotically stable. Similarly, if V1 (0) < V2 (0) < 0 then equilibrium x = 0, u 1 = 0 is locally asymptotically stable. References Courchamp F, Berec L, Gascoigne J (2008) Allee effects. Oxford University Press, Oxford Cressman R, Krivan V (2013) Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds. J Math Biol 67:329–358 Cressman R, Kˇrivan V (2006) Migration dynamics for the ideal free distribution. Am Nat 168:384–397 Cressman R, Kˇrivan V (2010) The ideal free distribution as an evolutionarily stable state in densitydependent population games. Oikos 119:1231–1242 Fretwell DS, Lucas HL (1969) On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoret 19:16–32 Fretwell SD (1972) Populations in a seasonal environment. Princeton University Press, Princeton Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge Kˇrivan V, Cressman R (2009) On evolutionary stability in prey-predator models with fast behavioral dynamics. Evol Ecol Res 11:227–251 Kˇrivan V, Cressman R, Schneider S (2008) The ideal free distribution: a review and synthesis of the gametheoretic perspective. Theoret Popul Biol 73:403–425 Morris DW (1988) Habitat-dependent population regulation and community structure. Evol Ecol 2:253–269 Morris DW (2002) Measuring the Allee effect: positive density dependence in small mammals. Ecology 83:14–20 Pohley JH, Thomas B (1983) Non-linear ESS models and frequency dependent selection. Biosystems 16:87–100 Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156 Tran T, Cressman R (2014) The Ideal Free Distribution and Evolutionary Stability in habitat selection games with linear fitness and Allee effect (Mimeo)
123
