Stability and Complexity in Prey Predators Models

Are more complex systems more stable? R. M. May used an extension of the Lotka-Volterra model to show that this is not the case. Let us consider a food web consisting of n prey and n generalist predators. Assuming that such a system has an equilibrium at which all species exist at positive densities it can be shown that the corresponding eigenvalues occur in pairs, each pair having the form £ + i) and — £ — i). Thus, there are two possibilities. Either real parts of all eigenvalues are zero in which case the equilibrium is neutrally stable exactly as in the case of the Lotka-Volterra prey-predator model [1]. If there exists an eigenvalue with a negative real part, then there must be also an eigenvalue with a positive real part which means that the equilibrium in n-prey-n-predator model is unstable. Thus, it is clear that the n-prey-n- predator system at best has the same stability property as the corresponding Lotka-Volterra prey-predator model. As the number of species increases it is more likely that among the eigenvalues there will be an eigenvalue with a positive real part and the equilibrium will be unstable. This and other models lead to prediction that complexity destabilizes food webs. These studies considered only the case where interaction strengths are fixed. In other words, they exclude the possibility of adaptive predator foraging behavior, or prey escape strategies. Recent studies show that when predator foraging behavior is adaptive (similarly as described in model [13]), the negative relation between food web complexity and community persistence can be reversed.

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