Analysis of a stage-structured predator-prey model with Crowley-Martin function

Abstract

Predator-prey interactions may have a large impact on the overall properties of a community. In this paper, we introduce an improved stage-structured predator-prey model with Crowley-Martin function. The possibility of existence of nonnegative equilibria has been considered. The locally asymptotic stability of nonnegative equilibria are considered and the permanence of the system are presented. By stability analysis we obtained sufficient conditions in the parameters for the global stability of the positive equilibrium. Numerical simulations are presented to illustrate the results.

Positive solutions for a modified Leslie–Gower prey–predator model with Crowley–Martin functional responses

“…where m, A and B are positive parameters that describe the effects of capture rate handling time and the magnitude of interference among predators, respectively, on the feeding rate (see [17,37,38,[50][51][52] for related works). By take above considerations, Ali and Jazar [2] consider a prey-predator model which incorporates a modified version of the Leslie type functional response parameters, in which c measures the magnitude of interference among prey.…”

“…Empirical evidence suggests that the spatial scale and structure of environment can influence population interactions [9]. By considering the above reasons, we consider the following reaction-diffusion equations: •ã denotes birth rate of prey, andb denotes the birth rate of predator; K is the carrying capacity of prey populationũ, e 2 represents intra-specific pressures of predator; • the termλṽ/[(1 +cũ)(1 +dṽ)] is known as Crowley-Martin interactions, in whichλ andd describe the effects of capture rate handling time andc describes the magnitude of interference among predators (see [17,37,38,[50][51][52]); • the termμṽ/(ũ+r) is a modified version of Leslie-Gower functional response, in whichμ is the maximum value which per capita reduction rate ofṽ can attain,r measure the extent to which environment provides protection to predatorṽ (see [5]). …”

Dynamical Behaviors of Stochastic Delayed One-Predator and Two-Competing-Prey Systems with Holling Type IV and Crowley-Martin Type Functional Responses

“…The dynamic relationship among species is a significant theme whether in ecology or in mathematical ecology because of its importance and universal existence with many concerned biological systems (see [1]). A lot of systems about predator-prey behaviors have been proposed (see [2][3][4]).…”

“…1 and 2 are the rates of conversion of nutrients into the production of predator. And all the parameters in system (2) are constants. It is very necessary to point out that 3 /(1 + 1 + + 1 ) is a special functional response; when = = = 0 it becomes a linear mass-action function response (or Holling type I functional response), when = = 0 it becomes a Holling type II functional response, when = 0 it becomes a modified Holling type II functional response, and when = it becomes a Crowley-Martin functional response.…”

Analysis of an Impulsive One-Predator and Two-Prey System with Stage-Structure and Generalized Functional Response

“…On the other hand, for predator-prey model, in description of the relationship between predator and prey, a crucial element is the classic definition of a predator's functional response. Recently, the dynamics of predator-prey systems with different kinds of functional responses have been studied in relevant literature, such as Holling type [6], Crowley-Martin type [7][8][9], Beddington-DeAngelis type [10,11], Watt type [12,13], and Ivlev type [14]. For example, Gakkhar and Naji [15] investigated the dynamical behaviors of the following threespecies system with nonlinear functional response: …”