THEORETICAL
POPULATION
Stability
BIOLOGY
7, 149-155 (1975)
of Ecosystems with Complex CLARK
Mathematics
Department,
University
Food Webs
JEFFRIES
of Saskatchewan,
Saskatoon S7N OWO, Canada
Received August 30, 1974
In a complex food web, various interaction pathways may connect pairs of species. A high carnivore might prey upon both a lesser carnivore and the prey of the lesser carnivore, forming a “loop ” in the food web. This report contains one technical definition for stability of an ecosystem. Sufficient conditions for stability are presented. The conditions require that loops in food webs be in a sense “balanced.”
According to modern ecological thinking, ecosystems are always evolving or developing. For example, following a fire there is the development of a new forest community, which might be quite orderly and predictable. Gradual climatic changes over long time spans might be accompanied by changes in species populations and nutrient concentrations. That is, on a long term basis the equilibrium state of an ecosystem changes. However, on a short term basis, such dynamic equilibrium must be stable dynamic equilibrium. There must be within the system short term interactions which keep ecosystem development on a steady course. In the short term in some communities, predation interactions may be a factor in the maintenance of stable dynamic equilibrium. This paper contains a set of sufficient conditions, a “strategy,” for stability with predation interactions. Suppose there are a number n of variables vu1, TJ~,..., V~ which, at the ecosystem level, describe the system. Each zli is assumed to be a differentiable function of time t, and together the variables constitute the state vector v. We assume the system is deterministic. That is, we assume that if the time (of year, since the last fire, or whatever) and the state vector are known, then the rate of change of the state vector and the future (or possible futures) of the state vector may be determined. That is, we assume first dvJdt
= Fi(v,
t),
(1)
where Fi(v, t) are some real-valued functions, and second that, given an initial state vector and time, there is at least one solution to the dynamical system (1). Suppose during the development of an ecosystem it is possible to describe 149 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
150
CLARK
JEFFRIES
the average or typical state vector, the development the system would follow without, say, human intervention. Let (si(t), sa(t),..., sJt>) = s(t) denote such a typical solution of (1). Hence s solves dsildt = Fi(s, t).
(2)
Following a limited disturbance, such as a brief period of human “harvesting” of some member species, the system should return, with time, to the typical solution s(t). For example, humans might every few decades catch most or all of the mature brook trout in a remote stream. Within a few years of such a harvest, the trout population and other populations would be expected to approximate natural levels. Let xi = TJ~- si denote the components of the difference vector x, a mathematical convenience. Define new functions Gi(x, t) as Gi(x, t) = Fi(v, t) - Fi(s, t). Thus we have for the difference vector dxJdt = Gi(x, t),
(3)
with each Gi(O, t) = 0. Hence if Xi(t) > 0, the ith variable is greater than typical for that time t, and so on. The typical state vector is represented by the constant vector 0 = (0,O ,..., 0). We wish to write the rate of change functions {Gi} for (3) in a form more susceptible to ecological interpretation. Let each of the difference variables xi be measured according to an “admissible” function fi , using the terminology of Lefschetz [I]. If fi is admissible, then f,(O) = 0 and otherwise fi(z) has the same sign as z. Also fi is required to be continuous with both
unbounded. We could, and a reader not interested in great generality might, simply choose each fi(z) = z. 0 r we might choose a fancier way to measure the variables, say, fi(z) = ez - 1. Or we might choose fi(xl) = x, , fi(x2) = e”e - 1, f3(x3) = (~a)~,and so on. Hence we decompose the functions Gi(x, t) so that dXi/dt = i
aii(x, t)fj(X,)v
?=I
(4)
Now each aij represents the rate of change in xi per fj-unit of xj brought about by xj at the particular state vector x and time t. The magnitudes of aij and uji reflect the strengths of the interactions between xi and xi . An additional interpretation of the interaction matrix follows. Suppose uij(x, t) > 0 for some x, t. Suppose all variables are typical (= 0) except xi .
STABILITY
OF ECOSYSTEMS WITH
COMPLEX FOOD WEBS
151
Then we have dxJdt > 0, that is, xi would begin to increase, possibly initiating other reactions in the system. Here xi could represent a predator of xj . An atypical abundance of xj initiates an increase in xi above typical. The terms aii are subject to a special interpretation. If aii(x, t) < 0 and all variables are typical except xi > 0, then initially xi will decrease. If aJx, t) < 0 and all variables are typical except xi < 0, then initially xi will increase. Such interactions might be called self-regulation or self-crowding interactions. So far the interaction matrix has only been discussed qualitatively.
STABILITY
CONDITIONS
For (4) suppose the interaction matrix functions {aij> are all continuous and the measure functions {fj} all admissible. Suppose in terms of initial conditions at least one solution state vector always exists for (4). Then we may quote the following. THEOREM. (i)
(ii)
Suppose
for each i, x, t we have u,,(x, t) < 0, for each i < j, x, t we have uij(x, t) = Kijuji(x,
t) where each Kij is
u
negative constant, and
(iii) for each set of three or more distinct indices i, j, k,..., p and each x, t we have / uijujlc ... UDi / = 1UiP *** ulcjuji I. Then 0 is a (neutral or asymptotic) stable stute,for (4). Condition (i) implies that for some x, t, uii may be zero, and for other x, t, uii may be negative. Or uii may be always zero or always negative. Condition (ii) implies that for each X, t, i # j either aii = uji = 0, or aij # 0 and uji # 0. Condition (iii) allows both products to be zero for some or all sets of three or more distinct indices and some or all x, t. Condition (iii) pertains only to loops of interactions between three or more variables, should any loops occur. It is important to note that the conditions are independent of resealing. That is, if we let yi = yixi for some scale factors yi , each positive, then we obtain a new dynamical system expressing dy,/dt in terms of y, t, namely,
dyildt= fl bdy, t)gj(Yj),
(5)
j=l
where b&, t) = yiaij(x, t) and gi(yi) = fi(xi). N ow each gj is admissible if and only if each fj is admissible. Also the new interaction matrix {bij} fulfills the conditions if and only if (aii} fulfills the conditions.
152
CLARK JEFFRIES
The biological interpretation of condition (i) is simply that any intraspecific or intravariable interactions be of the self-regulating or self-crowding type. The biological interpretation of condition (ii) is first that for i # j, aij # 0 implies aji # 0. In elementary language, wolves might in some seasons feed mostly on mice, not deer. If so, then wolves according to condition (ii) should not in such a season kill deer without feeding on deer. If for some i # j, x, t, aij # 0, then condition (ii) implies the constancy of the ratio aii/aji for “nearby” x and t. We offer the following interpretation. Suppose relative to a given predator and prey, a preferred prey suddenly becomes abundant. Then the per animal (actually per f,-unit or fj-unit) effects of the given predator and prey on each other will decrease, that is, both aij and uji will decrease in absolute value. Condition (ii) requires simply that these decreases be proportional.
DISCUSSION
OF THE BALANCED
LOOPS CONDITION
Suppose in a pond ecosystem phytoplankton x1 are preyed upon zooplankton xa and bloodworms xs . Suppose diptera larvae x4 prey upon zooplankton. Finally suppose sunfish x5 prey upon diptera larvae, zooplankton and bloodworms. This system may be graphically represented in Fig. 1.
FIGURE I
The directed line from xi to X, , for example, represents the assumption that for some x, t bloodworms prey upon phytoplankton and so a&x, t) < 0 and u&x, t) > 0. We also assume that the plankton variables x1 and x2 are selfregulating, denoted by the circular interaction lines. Hence we assume for at
STABILITY
OF ECOSYSTEMS
WITH
COMPLEX
FOOD
WEBS
153
least some x, t and jz, tl that a,,(~, t) < 0 and a,,($ i) < 0. A detailed biological justification for this assumption presumably citing competition for light, dissolved gases, or other factors, is beyond the scope of this paper. For more information see Odum [2]. We will only say that by self-regulation we imply that if all variables except one are artificially held at 0, and if the perturbed variable still returns to the zero equilibrium state (the typical state), then that one variable is self-regulating. It may be that using this definition of selfregulation, most variables may be assumed self-regulating. At any rate, in the pond ecosystem outlined above, an approximation of condition (iii) might be part of a strategy for stability. Condition (iii), the balanced loop contition, is met if
Note that the balancing of the 2, 4, 5 and 1, 2, 3, 5 loops implies the balancing of the 1, 2, 3, 4, 5 loop. In fact it is generally true that the balancing of all the “small” loops in a system implies the balancing of all the “large” loops. Now / u25/u52/ may be thought of as a sort of “cost-benefit ratio” to the system for the zooplankton and sunfish interaction, and similarly for / ua4/u4zj, I Q/+,~ 1, and so on. We may think of the variables {xi} as representing not the numbers of individuals but the energy tied up as biomass in the ith form or species compartment. Thus there is an interesting interpretation of condition (iii). Namely, in a food web with various trophic levels, distinct paths which begin at one species, say, a producer, and end at one species, say, a top carnivore, should tend to be approximately equal in terms of the products of the cost-benefit ratios of the predation links involved. Different energy pathways should have about the same energy cost-benefit ratio products. It should be noted that in a model of the pond ecosystem the interaction coefficients would be highly variable, depending upon season. A strong feature of the above conditions is that such variability is in itself consistent with the conditions.
REMARKS
May [3] and Jeffries [4] have written of first-order or linear models without loops of predation links. Asymptotic stability, stronger than neutral stability, is obtained for such models when self-regulating species occur at certain locations in the food web. A similar result could be formulated for the model in this paper. In particular, if conditions (i), ( ii ) , and (iii) are met and if all uii(x, t) < E < 0, then the system (4) is “uniformly asymptotically stable in the large,” as defined in Willems [5].
154
CLARK
JEFFRIES
As the referee for this paper has pointed out, if each ~(x, t) < 0 and each ] aii(x, t)l is “much larger” than any / aij(x, t)l, i # j, then stability may be also obtained, regardless of the signs, ratios, and so on of the terms aii , i # j. See, for one result in this direction, Siljak [6]. Recall that by self-regulation we imply that if all variables are driven at equilibrium except one self-regulating variable, then that one variable will nonetheless stably approach equilibrium. Possibly among plants, where predation links are not obviously abundant to me, the strong self-regulation strategy is adopted. Possibly among some animals the balanced loop strategy is an important component of a strategy for dynamic equilibrium stability.
APPENDIX
Proof that conditions (i), (ii), and (iii) guarantee the neutral or asymptotic stability of (4) and the boundedness of all trajectories for (4) in terms of initial conditions: We will define n positive real numbers {&) satisfying the relation hiaii = --Xjuj, for all x, t using the following scheme. Let h, = 1. Let {i, j,...} be the set of all indices of variables which interact with x1 . (A variable xi interacts with x1 if there is some x, t such that an # 0, or alternatively, a, # 0.) If the set of such indices is empty, start over with x2 and ha = 1. Otherwise, for each such index i let &a, = --a,, . Next consider the (possibly empty) set of all indices {K, Z,...} of variables which interact with at least one of {xi , xj ,...}. Define for new indices {h, , h, ,.,.} by h,a,i = --hiaik . Condition (iii) implies that this scheme is consistent. For suppose xa and xs interact with x1 and with each other. Then we must have &a,, = --h,a,, and h,u,, = --Alal . Also we have h,a,, = --h,a,, , p recisely because condition (iii) implies 1a12u23u31/a13u32u21 / = 1. So it goes for all loops of three or more interacting variables. That is, this scheme may be extended to those variables which interact with {xlc , xI ,...}, and so on. When a maximal set of variables containing x, is so obtained, start over with some other variable x, , setting h, = 1, if any variables remain. The final result is a set of positive numbers {hi) such that if i fj and xi and xj interact, then X,aij = -Ajaji for x, t such that aii # 0, aji # 0, and so for all x, t. If i # j and xi and xi do not interact, then Xiaij = --hi+ nonetheless, of course. So for all i # j, x, t we have Xiaij(x, t) = --h+ji(x, t), that is, h~‘2a&~1J2= -h:‘2a,,X;1’2. To summarize this paragraph we may say that the conditions are sufficient to guarantee the existence of n positive constants (hi} such that the similarity transformation on {uij} by the diagonal matrix with h$‘2in the i, i-entry renders {au} into skew form plus nonpositive diagonal entries for each x, t.
STABILITY
OF ECOSYSTEMS WITH
COMPLEX FOOD WEBS
155
Next define a Liapunov function v(x) as
p(x) =Chjjozi fi(z) dz. Clearly q(x) > 0 unless x = 0. Clearly v(x) strictly increases without bound on all radial lines in n-space away from 0. Finally, dyldt
= f
A
POPULATION
Stability
BIOLOGY
7, 149-155 (1975)
of Ecosystems with Complex CLARK
Mathematics
Department,
University
Food Webs
JEFFRIES
of Saskatchewan,
Saskatoon S7N OWO, Canada
Received August 30, 1974
In a complex food web, various interaction pathways may connect pairs of species. A high carnivore might prey upon both a lesser carnivore and the prey of the lesser carnivore, forming a “loop ” in the food web. This report contains one technical definition for stability of an ecosystem. Sufficient conditions for stability are presented. The conditions require that loops in food webs be in a sense “balanced.”
According to modern ecological thinking, ecosystems are always evolving or developing. For example, following a fire there is the development of a new forest community, which might be quite orderly and predictable. Gradual climatic changes over long time spans might be accompanied by changes in species populations and nutrient concentrations. That is, on a long term basis the equilibrium state of an ecosystem changes. However, on a short term basis, such dynamic equilibrium must be stable dynamic equilibrium. There must be within the system short term interactions which keep ecosystem development on a steady course. In the short term in some communities, predation interactions may be a factor in the maintenance of stable dynamic equilibrium. This paper contains a set of sufficient conditions, a “strategy,” for stability with predation interactions. Suppose there are a number n of variables vu1, TJ~,..., V~ which, at the ecosystem level, describe the system. Each zli is assumed to be a differentiable function of time t, and together the variables constitute the state vector v. We assume the system is deterministic. That is, we assume that if the time (of year, since the last fire, or whatever) and the state vector are known, then the rate of change of the state vector and the future (or possible futures) of the state vector may be determined. That is, we assume first dvJdt
= Fi(v,
t),
(1)
where Fi(v, t) are some real-valued functions, and second that, given an initial state vector and time, there is at least one solution to the dynamical system (1). Suppose during the development of an ecosystem it is possible to describe 149 Copyright All rights
0 1975 by Academic Press, Inc. of reproduction in any form reserved.
150
CLARK
JEFFRIES
the average or typical state vector, the development the system would follow without, say, human intervention. Let (si(t), sa(t),..., sJt>) = s(t) denote such a typical solution of (1). Hence s solves dsildt = Fi(s, t).
(2)
Following a limited disturbance, such as a brief period of human “harvesting” of some member species, the system should return, with time, to the typical solution s(t). For example, humans might every few decades catch most or all of the mature brook trout in a remote stream. Within a few years of such a harvest, the trout population and other populations would be expected to approximate natural levels. Let xi = TJ~- si denote the components of the difference vector x, a mathematical convenience. Define new functions Gi(x, t) as Gi(x, t) = Fi(v, t) - Fi(s, t). Thus we have for the difference vector dxJdt = Gi(x, t),
(3)
with each Gi(O, t) = 0. Hence if Xi(t) > 0, the ith variable is greater than typical for that time t, and so on. The typical state vector is represented by the constant vector 0 = (0,O ,..., 0). We wish to write the rate of change functions {Gi} for (3) in a form more susceptible to ecological interpretation. Let each of the difference variables xi be measured according to an “admissible” function fi , using the terminology of Lefschetz [I]. If fi is admissible, then f,(O) = 0 and otherwise fi(z) has the same sign as z. Also fi is required to be continuous with both
unbounded. We could, and a reader not interested in great generality might, simply choose each fi(z) = z. 0 r we might choose a fancier way to measure the variables, say, fi(z) = ez - 1. Or we might choose fi(xl) = x, , fi(x2) = e”e - 1, f3(x3) = (~a)~,and so on. Hence we decompose the functions Gi(x, t) so that dXi/dt = i
aii(x, t)fj(X,)v
?=I
(4)
Now each aij represents the rate of change in xi per fj-unit of xj brought about by xj at the particular state vector x and time t. The magnitudes of aij and uji reflect the strengths of the interactions between xi and xi . An additional interpretation of the interaction matrix follows. Suppose uij(x, t) > 0 for some x, t. Suppose all variables are typical (= 0) except xi .
STABILITY
OF ECOSYSTEMS WITH
COMPLEX FOOD WEBS
151
Then we have dxJdt > 0, that is, xi would begin to increase, possibly initiating other reactions in the system. Here xi could represent a predator of xj . An atypical abundance of xj initiates an increase in xi above typical. The terms aii are subject to a special interpretation. If aii(x, t) < 0 and all variables are typical except xi > 0, then initially xi will decrease. If aJx, t) < 0 and all variables are typical except xi < 0, then initially xi will increase. Such interactions might be called self-regulation or self-crowding interactions. So far the interaction matrix has only been discussed qualitatively.
STABILITY
CONDITIONS
For (4) suppose the interaction matrix functions {aij> are all continuous and the measure functions {fj} all admissible. Suppose in terms of initial conditions at least one solution state vector always exists for (4). Then we may quote the following. THEOREM. (i)
(ii)
Suppose
for each i, x, t we have u,,(x, t) < 0, for each i < j, x, t we have uij(x, t) = Kijuji(x,
t) where each Kij is
u
negative constant, and
(iii) for each set of three or more distinct indices i, j, k,..., p and each x, t we have / uijujlc ... UDi / = 1UiP *** ulcjuji I. Then 0 is a (neutral or asymptotic) stable stute,for (4). Condition (i) implies that for some x, t, uii may be zero, and for other x, t, uii may be negative. Or uii may be always zero or always negative. Condition (ii) implies that for each X, t, i # j either aii = uji = 0, or aij # 0 and uji # 0. Condition (iii) allows both products to be zero for some or all sets of three or more distinct indices and some or all x, t. Condition (iii) pertains only to loops of interactions between three or more variables, should any loops occur. It is important to note that the conditions are independent of resealing. That is, if we let yi = yixi for some scale factors yi , each positive, then we obtain a new dynamical system expressing dy,/dt in terms of y, t, namely,
dyildt= fl bdy, t)gj(Yj),
(5)
j=l
where b&, t) = yiaij(x, t) and gi(yi) = fi(xi). N ow each gj is admissible if and only if each fj is admissible. Also the new interaction matrix {bij} fulfills the conditions if and only if (aii} fulfills the conditions.
152
CLARK JEFFRIES
The biological interpretation of condition (i) is simply that any intraspecific or intravariable interactions be of the self-regulating or self-crowding type. The biological interpretation of condition (ii) is first that for i # j, aij # 0 implies aji # 0. In elementary language, wolves might in some seasons feed mostly on mice, not deer. If so, then wolves according to condition (ii) should not in such a season kill deer without feeding on deer. If for some i # j, x, t, aij # 0, then condition (ii) implies the constancy of the ratio aii/aji for “nearby” x and t. We offer the following interpretation. Suppose relative to a given predator and prey, a preferred prey suddenly becomes abundant. Then the per animal (actually per f,-unit or fj-unit) effects of the given predator and prey on each other will decrease, that is, both aij and uji will decrease in absolute value. Condition (ii) requires simply that these decreases be proportional.
DISCUSSION
OF THE BALANCED
LOOPS CONDITION
Suppose in a pond ecosystem phytoplankton x1 are preyed upon zooplankton xa and bloodworms xs . Suppose diptera larvae x4 prey upon zooplankton. Finally suppose sunfish x5 prey upon diptera larvae, zooplankton and bloodworms. This system may be graphically represented in Fig. 1.
FIGURE I
The directed line from xi to X, , for example, represents the assumption that for some x, t bloodworms prey upon phytoplankton and so a&x, t) < 0 and u&x, t) > 0. We also assume that the plankton variables x1 and x2 are selfregulating, denoted by the circular interaction lines. Hence we assume for at
STABILITY
OF ECOSYSTEMS
WITH
COMPLEX
FOOD
WEBS
153
least some x, t and jz, tl that a,,(~, t) < 0 and a,,($ i) < 0. A detailed biological justification for this assumption presumably citing competition for light, dissolved gases, or other factors, is beyond the scope of this paper. For more information see Odum [2]. We will only say that by self-regulation we imply that if all variables except one are artificially held at 0, and if the perturbed variable still returns to the zero equilibrium state (the typical state), then that one variable is self-regulating. It may be that using this definition of selfregulation, most variables may be assumed self-regulating. At any rate, in the pond ecosystem outlined above, an approximation of condition (iii) might be part of a strategy for stability. Condition (iii), the balanced loop contition, is met if
Note that the balancing of the 2, 4, 5 and 1, 2, 3, 5 loops implies the balancing of the 1, 2, 3, 4, 5 loop. In fact it is generally true that the balancing of all the “small” loops in a system implies the balancing of all the “large” loops. Now / u25/u52/ may be thought of as a sort of “cost-benefit ratio” to the system for the zooplankton and sunfish interaction, and similarly for / ua4/u4zj, I Q/+,~ 1, and so on. We may think of the variables {xi} as representing not the numbers of individuals but the energy tied up as biomass in the ith form or species compartment. Thus there is an interesting interpretation of condition (iii). Namely, in a food web with various trophic levels, distinct paths which begin at one species, say, a producer, and end at one species, say, a top carnivore, should tend to be approximately equal in terms of the products of the cost-benefit ratios of the predation links involved. Different energy pathways should have about the same energy cost-benefit ratio products. It should be noted that in a model of the pond ecosystem the interaction coefficients would be highly variable, depending upon season. A strong feature of the above conditions is that such variability is in itself consistent with the conditions.
REMARKS
May [3] and Jeffries [4] have written of first-order or linear models without loops of predation links. Asymptotic stability, stronger than neutral stability, is obtained for such models when self-regulating species occur at certain locations in the food web. A similar result could be formulated for the model in this paper. In particular, if conditions (i), ( ii ) , and (iii) are met and if all uii(x, t) < E < 0, then the system (4) is “uniformly asymptotically stable in the large,” as defined in Willems [5].
154
CLARK
JEFFRIES
As the referee for this paper has pointed out, if each ~(x, t) < 0 and each ] aii(x, t)l is “much larger” than any / aij(x, t)l, i # j, then stability may be also obtained, regardless of the signs, ratios, and so on of the terms aii , i # j. See, for one result in this direction, Siljak [6]. Recall that by self-regulation we imply that if all variables are driven at equilibrium except one self-regulating variable, then that one variable will nonetheless stably approach equilibrium. Possibly among plants, where predation links are not obviously abundant to me, the strong self-regulation strategy is adopted. Possibly among some animals the balanced loop strategy is an important component of a strategy for dynamic equilibrium stability.
APPENDIX
Proof that conditions (i), (ii), and (iii) guarantee the neutral or asymptotic stability of (4) and the boundedness of all trajectories for (4) in terms of initial conditions: We will define n positive real numbers {&) satisfying the relation hiaii = --Xjuj, for all x, t using the following scheme. Let h, = 1. Let {i, j,...} be the set of all indices of variables which interact with x1 . (A variable xi interacts with x1 if there is some x, t such that an # 0, or alternatively, a, # 0.) If the set of such indices is empty, start over with x2 and ha = 1. Otherwise, for each such index i let &a, = --a,, . Next consider the (possibly empty) set of all indices {K, Z,...} of variables which interact with at least one of {xi , xj ,...}. Define for new indices {h, , h, ,.,.} by h,a,i = --hiaik . Condition (iii) implies that this scheme is consistent. For suppose xa and xs interact with x1 and with each other. Then we must have &a,, = --h,a,, and h,u,, = --Alal . Also we have h,a,, = --h,a,, , p recisely because condition (iii) implies 1a12u23u31/a13u32u21 / = 1. So it goes for all loops of three or more interacting variables. That is, this scheme may be extended to those variables which interact with {xlc , xI ,...}, and so on. When a maximal set of variables containing x, is so obtained, start over with some other variable x, , setting h, = 1, if any variables remain. The final result is a set of positive numbers {hi) such that if i fj and xi and xj interact, then X,aij = -Ajaji for x, t such that aii # 0, aji # 0, and so for all x, t. If i # j and xi and xi do not interact, then Xiaij = --hi+ nonetheless, of course. So for all i # j, x, t we have Xiaij(x, t) = --h+ji(x, t), that is, h~‘2a&~1J2= -h:‘2a,,X;1’2. To summarize this paragraph we may say that the conditions are sufficient to guarantee the existence of n positive constants (hi} such that the similarity transformation on {uij} by the diagonal matrix with h$‘2in the i, i-entry renders {au} into skew form plus nonpositive diagonal entries for each x, t.
STABILITY
OF ECOSYSTEMS WITH
COMPLEX FOOD WEBS
155
Next define a Liapunov function v(x) as
p(x) =Chjjozi fi(z) dz. Clearly q(x) > 0 unless x = 0. Clearly v(x) strictly increases without bound on all radial lines in n-space away from 0. Finally, dyldt
= f
A
