The effect of refuge and immigration in a predator–prey system in the presence of a competitor for the prey

This paper considers the effect of immigration and refuge on the dynamics of a three species system in which one predator feeds on one of two competing species. Immigration is assumed only for the species which is not attacked by the predator.The main results address the stability of the system. Namely, it is shown that increasing the number of refuges stabilizes the system, whereas the opposite holds true by increasing the immigration rate. Also, one result about the persistence of the system and one concerning the global stability of the coexistence equilibrium are presented. Some numerical simulations illustrate the obtained results.     

具有避难所和Ivlev功能性反应的捕食系统的稳定性分析

本文研究了一类具有Ivlev功能性反应的捕食系统,其中食饵种群具有避难所．本文的目的是对模型进行系统的分析,并讨论由于生物体之间相互作用可能产生的一些有意义的定性结果．     

Existence of positive solutions for a prey-predator model with refuge and diffusion

We are concerned with a prey-predator reaction-diffusion model with monotonic functional response and specific refuge size. We discuss both existence and nonexistence of positive solutions of the model by using fixed point index theory and bifurcation theory as the main argument tools. We also discuss bifurcation solutions of which stability is established by using spectrum analysis methods. Moreover, we analyze the effects of refuge size on the dynamics of the model.     

Paradox of enrichment: A fractional differential approach with memory

The paradox of enrichment (PoE) proposed by Rosenzweig [M. Rosenzweig, The paradox of enrichment, Science 171 (1971) 385–387] is still a fundamental problem in ecology. Most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking. Knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels. PoE resolution via these two organizational levels can be interpreted as a microscopic- and macroscopic-level solution. Fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes. The derivatives can be physically interpreted via two time scales that are considered simultaneously: the ideal, equably flowing homogeneous local time, and the cosmic (inhomogeneous) non-local time. Several mechanisms and theories have been proposed to resolve the PoE problem, but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non-local effects, which capture memory. Here we formulate the fractional counterpart of the Rosenzweig model and analyze the stability behavior of a system. We conclude that there is a threshold for the memory effect parameter beyond which the Rosenzweig model is stable and may be used as a potential agent to resolve PoE from a new perspective via fractional differential equations.     

Effects of prey refuge on a ratio-dependent predator–prey model with stage-structure of prey population

In this paper, a stage-structured predator–prey model is proposed and analyzed to study how the type of refuges used by prey population influences the dynamic behavior of the model. Two types of refuges: those that protect a fixed number of prey and those that protect a constant proportion of prey are considered. Mathematical analyses with regard to positivity, boundedness, equilibria and their stabilities, and bifurcation are carried out. Persistence condition which brings out the useful relationship between prey refuge parameter and maturation time delay is established. Comparing the conclusions obtained from analyzing properties of two types of refuges using by prey, we observe that value of maturation time at which the prey population and hence predator population go extinct is greater in case of refuges which protect a constant proportion of prey.