Dynamics of a three species food chain model with Crowley–Martin type functional response

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]). …”

Controlling Chaotic Dynamics of a Continuous Ecological Model

“…Since the work of May [22] in the last century, researchers in ecology, biology and applied engineering reported the existence of chaos for different range of parameter value of the system, see [2,8,19,20,29]. The principle of competitive exclusion is one of the best known result in theoretical ecology.…”

“…Ayala [10] proved experimentally that two species of Drosophila could coexist upon a single limit resource. Ayala's experiment leads to a large body of work, which continuous to this day, see [1,2,6,8,[15][16][17][19][20][21][22][23][24][25]. Upadhyay [20] studied Rosenzweig-McArthur [18] model to investigate the existence of complex dynamical behavior in a food chain model involving Holling type IV functional response of the form wx d+x+x 2 /i and the effects of the intraspecific competition in the first level of the model is also studied.…”

“…Ayala's experiment leads to a large body of work, which continuous to this day, see [1,2,6,8,[15][16][17][19][20][21][22][23][24][25]. Upadhyay [20] studied Rosenzweig-McArthur [18] model to investigate the existence of complex dynamical behavior in a food chain model involving Holling type IV functional response of the form wx d+x+x 2 /i and the effects of the intraspecific competition in the first level of the model is also studied. Sokol and Howell [28] suggested a simplified Holling type IV function of the form wx d+x 2 and found that it is simpler and better than the original function of Holling type IV, about how these functional responses obtained see [7,13,21,22] and [26][27][28].…”