J. theor. Biol. (1979) 79, 243-246
LETTERS TO THE EDITOR
A Generalized Prey-Predator
The evolution of a community of population consisting of several interacting species may be described by a set of non-linear equations, a well-known example being Lotka-Volterra model. The two-species L-V system is characterized by the property that it is conservative (Goel et al., 1971) and it possesses centre type of critical point. Though L-V model occupies an important position in the population dynamics of interacting species, it has some unrealistic features e.g. the oscillations are initial-value dependent and this model does not exhibit a limit cycle behaviour. We have investigated a modification of the L-V model wherein the interaction term is taken to be proportional to some power of the population sizes of the two species. This system is non-conservative and does not possess any centre type of critical point. We have observed, however, that under certam conditions this model can show a limit cycle behaviour. It is further seen that when the strength of the self-limiting growth term exceeds a critical value, the limit cycle behaviour is not observed at all. For a community comprising of one prey and one predator with populations N, and N, respectively, we write the evolution equation as
dN, __ = - yN, + /I2 N’fN; dt
both m and n being positive numbers. By means of appropriate changes of variables the equations (1) may be written as i = ax(l-/!?x)-.xrny” = Xh.i(x,y) (2) p = -y+x”y” = YKZ(X, 4’) I where x and y are the variables corresponding to prey and predator respectively and the dots represent (new) time-derivatives. In order that these equations may be biologically plausible, a set of conditions, known as Kolmogorov conditions (Kolmogorov, 1936; May, 1972) must be satisfied. This requires that n < 1 and (m+n)> 1. 243 0022-5193/79/140243+04
t‘j 1979 .Academic
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The singular point (x,,y,) following equations
of the system can be determined
Yo = axo(l
The singular point (0,O) is excluded because it is a saddle point. We introduce the parameters E, 6, n, v and p defined by & = LY(l-m),
6 = (1 -n),
q = a(m+n-
With the help of usual linearization procedure one can show that the singular point will be a focus if (E+ v - 6)’ < 4(n - v) and it will be a stable focus or an unstable focus according as (E + v) < 6 or > 6. Now, Kolmogorov’s theorem (May, 1972) leads to the result that when Kolmogorov conditions are satisfied, the singular point is surrounded by a stable limit cycle provided it is an unstable focus. It is obvious that the region A in m-n space, which lies inside the parabola (E+ v- 6)’ = 4(n- v) and above the straight line E+V = 6, corresponds to unstable focus. We must note that in m-n space Kolmogorov conditions impose two other restrictions stated earlier, namely n < 1 and (m + n) > 1. So we get another region B lying below the line n = 1 and above the line m+n = 1, noting that both m and n are always positive. We have seen that only when the value of p is less than a critical value there exists an overlap between the regions A and B. This overlap is the domain of limit cycle. The main feature of the prey-predator system governed by equations (1) is the existence of limit cycle for certain values of m and n. It is observed that the domain of limit cycle is very much altered by the magnitudes of the parameters a and ~1. Further we see that for a fixed value of cc,the domain of limit cycle diminishes as p increases. The existence of limit cycle can be envisaged geometrically as illustrated by Rescigno & Richardson (1973). It is evident that the phase space is partitioned into four regions by the curves ~r(x, y) = 0 and x2(x, y) = 0. One can show, by considering the signs of dy/dx in the four regions, that a phase space trajectory starting from infinity gradually spirals inward. On the other hand, for m < 1, n < 1 and (m + n) > 1, the point of intersection of the curves rcr(x, y) = 0 and K~(x, y) = 0 is an unstable focus. Consequently, a trajectory in the neighbourhood of the point of intersection will spiral outward (Fig. 1). Hence the existence of a stable limit cycle is evident.
FIG 1. The behaviour of the phase trajectories, when the critical point is an unstable focus surrounded by a stable limit cycle (m < 1, n s: 1, (m +n) > 1), is shown.
Finally, we may add another interesting point about this model. When b in equatxon (2) is zero, then by a transformation u = log (x” - ly”) v = log(x”y”-‘) equation
(2) can be written as ti = a, e”+b, e”+c, C = a2 e”+b, er+c2 I
These are essentially the Verhulst form of equations occurring in population dynamics. As shown by Trubatch & Franc0 (1974) one can construct a linear Lagrangian appropriate to these equations and apply variational method to obtain the frequency of oscillation. Depar.tment of Physics Ramsuday College, Amta Howroh, West Bengal, India Department of Physics Vidyasagar Evening College, Calcutta. India
Saha Institute of Nuclear Physics 92, Acharya Prafulla Chandra Road Calcutta-700009, India (Received
GOEL, N. S., MAITRA, S. C. & MONTROLL, E. W. (1971). Rev. mod. Phy. 43, 231. KOLMOGOROV, A. N. (1936). Giorn Ins&. Ital. degli Atruari 14, 1. MAY, R. M. (1972). Science 177,900. RESCIGNO, A. & RICHARDSON, I. W. (1973). In Foundations qf Mathematical Biolog-v, Ch. 4, (R. Rosen, ed.). London : Academic Press. TRUBATCH, S. L. & FRANCO, A. (1974). J. theor. Biol. 48, 299.