Complex dynamics of a discrete-time predator-prey system with Holling IV functional response
The paper deals with the dynamics of a discrete predator-prey system with the response function of Holling IV type, that is, a non-monotonic function with a hump. As the authors explain, to exclude existence of unrealistic negative solutions in the system, they start with a continuous system \[ \begin{aligned} \dot{x} & =rx\left( 1-\frac{x}{K}\right) -\frac{mxy}{ax^{2}+bx+1},\\ \dot{y} & =y\left( \frac{cx}{ax^{2}+bx+1}-d\right).\end{aligned} \] By applying a technique from [\textit{M. Fan} and \textit{K. Wang}, Math. Comput. Modelling 35, No. 9--10, 951--961 (2002; Zbl 1050.39022)], a system of differential equations with piecewise constant arguments is derived under the assumption that the average growth rates change regularly. The integration of a modified system over the interval \([n,n+1)\) yields the following discrete system \[ \begin{aligned} x\left( n+1\right) & =x\left( n\right) \exp\left[ r\left( 1-\frac{x\left( n\right) }{K}\right) -\frac{my\left( n\right) }{ax^{2}\left( n\right) +bx\left( n\right) +1}\right] ,\\ y\left( n+1\right) & =y\left( n\right) \exp\left[ \frac{cmx\left(n\right) }{ax^{2}\left( n\right) +bx\left( n\right) +1}-d\right] .\end{aligned}\tag{1} \] Here \(x\) and \(y\) stand for population densities of the prey and predator, respectively, \(K\) is the carrying capacity of the prey population, \(d\) is the death rate of predator population, \(c\) is a conversion rate of eaten prey into new predator abundance. Considering \(K\) as a bifurcation parameter, the authors demonstrate that the system (1) may undergo a flip bifurcation and a Neimark-Sacker bifurcation (a discrete analogue of the Andronov-Hopf bifurcation in continuous systems). Numerical simulations in Section 4 illustrate the theoretical results and reveal the complex dynamics of system (1) including orbits of period \(n,\) invariant cycles, period doubling bifurcations, and quasi-periodic orbits. The paper concludes with a brief discussion of the results.
scientific article; zbMATH DE number 6683354
scientific article