Abstract
In this article, we derive and analyze a novel predator-prey model with account for maturation delay in predators, ratio dependence, and Holling type III functional response. The analysis of the system's steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state. Demographic stochasticity in the model is explored by means of deriving a delayed chemical master equation. Using system size expansion, we study the structure of stochastic oscillations around the deterministically stable coexistence state by analyzing the dependence of variance and coherence of stochastic oscillations on system parameters. Numerical simulations of the stochastic model are performed to illustrate stochastic amplification, where individual stochastic realizations can exhibit sustained oscillations in the case, where deterministically the system approaches a stable steady state. These results provide a framework for studying realistic predator-prey systems with Holling type III functional response in the presence of stochasticity, where an important role is played by non-negligible predator maturation delay.
Highlights
Ever since the pioneering work of Lotka[1] and Volterra,[2] mathematical models of predator–prey type have provided tremendous insights into the dynamics of interactions between different species or, more widely, between interacting agents that have found application in biology and in a diversity of other areas, from immunology to economics.[3–6] The starting point for many of these models in ecological context is a general predator–prey model of Gause–Kolmogorov type,[7–9] u = uf(u) − vg (u, v), v = bvg (u, v) − dv, where u(t) and v(t) are abundances or population densities of prey and predator, respectively; f(u) describes the intrinsic per-capita growth rate of prey in the absence of predator; and d is the predator’s natural death rate
The analysis of the system’s steady states reveals conditions on predation rate, predator growth rate, and maturation time that can result in a prey-only equilibrium or facilitate simultaneous survival of prey and predators in the form of a stable coexistence steady state, or sustain periodic oscillations around this state
Motivated by recent work on plant diseases based on plant–insect interactions, in which insects are the predators feeding on plants playing the role of food source, we have proposed a new predator–prey model with ratio dependence and a Holling type III functional response
Summary
We will instead consider the following form of the Holling type III functional response: g(z) = ae−α/z, which satisfies the conditions of g(0) = 0, is monotonically increasing, and settles at a constant value as z → ∞. This functional response is reminiscent of the Ivlev (Holling II) trophic function g(z) = a 1 − e−αz 39,40 and of the Ricker model[41] for single-species populations.
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