A Predator–Prey Model with Ivlev's Functional Response

Zhang

et al. 2010

Ecological Modelling

In this paper, we investigate the emergence of a predator-prey system with Ivlevtype functional response and reaction-diffusion. We study how diffusion affects the stability of predator-prey coexistence equilibrium and derive the conditions for Hopf and Turing bifurcation in the spatial domain. Based on the bifurcation analysis, we give the spatial pattern formation, the evolution process of the system near the coexistence equilibrium point, via numerical simulation. We find that pure Hopf instability leads to the formation of spiral patterns and pure Turing instability destroys the spiral pattern and leads to the formation of chaotic spatial pattern. Furthermore, we perform three categories of initial perturbations which predators are introduced in a small domain to the coexistence equilibrium point to illustrate the emergence of spatiotemporal patterns, we also find that in the beginning of evolution of the spatial pattern, the special initial conditions have an effect on the formation of spatial patterns, though the effect is less and less with the more and more iterations. This indicates that for prey-dependent type predator-prey model, pattern formations do depend on the initial conditions, while for predator-dependent type they do not. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal dynamics.

Section: Introductionmentioning

confidence: 99%

Pattern formation of a predator–prey system with Ivlev-type functional response

Zhang

et al. 2010

Ecological Modelling

Mathematical Problems in Engineering

“…There are many works to investigate predator-prey system with Ivlev-type functional response. The results indicated that predator-prey systems with Ivlev-type have wide applicabilities in ecology, such as dynamics in predator-prey system [21][22][23][24][25], host-parasitoid system [26,27], and animal coat pattern [28].…”

Section: Introductionmentioning

confidence: 99%

Complex Dynamics of a Discrete-Time Predator-Prey System with Ivlev Functional Response

Baek

2018

The dynamics of a discrete-time predator-prey system with Ivlev functional response is investigated in this paper. The conditions of existence for flip bifurcation and Hopf bifurcation in the interior of R 2 + are derived by using the center manifold theorem and bifurcation theory. Numerical simulations are presented not only to substantiate our theoretical results but also to illustrate the complex dynamical behaviors of the system such as attracting invariant circles, periodic-doubling bifurcation leading to chaos, and periodic-halving phenomena. In addition, the maximum Lyapunov exponents are numerically calculated to confirm the dynamical complexity of the system. Finally, we compare the system to discrete systems with Holling-type functional response with respect to dynamical behaviors.

“…It naturally has been studied by a number of authors; for example, results can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,20,21,22] and the references cited therein. In the literature it is assumed that the rates r(u), q (u) respectively.…”

mentioning

confidence: 99%

“…We call E * the positive equilibrium hereafter. The main subjects of system (1.1) are the existence of a unique limit cycle which surrounds the positive equilibrium E * (see [1,9,10,11,13,16,17,21,22]) and the global asymptotic stability of the positive equilibrium E * (refer to [2,6,12,20]). The positive equilibrium E * is said to be globally asymptotically stable if E * is stable and if every orbit tends to E * .…”

mentioning

confidence: 99%

Homoclinic orbits in predator-prey systems with a nonsmooth prey growth rate

Sugie

Kimoto

2006

Quart. Appl. Math.

Abstract. This paper deals with Gause-type predator-prey models with a nonsmooth prey growth rate. Our models have a unique positive equilibrium and are under the influence of an Allee effect. A necessary and sufficient condition is given for the existence of homoclinic orbits whose α-and ω-limit sets are the positive equilibrium. The argument used here is based on some results of a system of Liénard type. The relation between homoclinic orbits and the Allee effect is clarified. A simple example is included to illustrate the main result. Some global phase portraits are also attached.