A predator–prey model with Crowley–Martin functional response: A nonautonomous study

Abstract

We investigate a nonautonomous predator–prey model system with a Crowley–Martin functional response. We perform rigorous mathematical analysis and obtain conditions for (a) global attractivity and permanence in the form of integrals which improve the traditional conditions obtained by using bounds of involved parameters; and (b) the existence of periodic solutions applying continuation theorem from coincidence degree theory which has stronger results than using Brouwer fixed point theorem. Our result also indicates that the global attractivity of periodic solution is positively affected by the predator's density dependent death rate. We employ partial rank correlation coefficient method to focus on how the output of the model system analysis is influenced by variations in a particular parameter disregarding the uncertainty over the remaining parameters. We discuss the relations between results (permanence and global attractivity) for autonomous and nonautonomous systems to get insights on the effects of time‐dependent parameters. Recommendations for Resource Managers: The natural environment fluctuates because of several factors, for example, mating habits, food supplies, seasonal effects of weathers, harvesting, death rates, birth rates, and other important population rates. The temporal fluctuations in physical environment (periodicity) plays a major role in community and population dynamics along with the impacts of population densities. Periodic system may suppress the permanence of its corresponding autonomous system with parameters being the averages of periodic parameters. As the human needs crosses a threshold level, then we require to observe the sustainability of resources of the associated exploited system. Therefore, the concept of stability and permanence become our main concern in an exploited model system (system with harvesting). The mutual interference at high prey density may leave negative effect on the permanence of the system. In harvested system, permanence becomes an important issue because if we harvest too many individuals then species may be driven to extinction. Interestingly, in many biological/agricultural systems, harvesting (due to fishing in marine system, hunting or disease) of a particular species/crop can only be more beneficial at certain times (e.g., the time and stage of harvest of a particular crop play greater role in its production and hence the particular crop is many times harvested at its physiological maturity or at harvest maturity).

Dynamic Behaviors of an Obligate Commensal Symbiosis Model with Crowley–Martin Functional Responses

“…In 1989, Crowley and Martin [43] proposed a functional response, which is similar to the Beddington-DeAngelis response function, but it includes one more term explaining mutual interferences of predators at the high density of its prey. Many scholars performed works on predator prey system with Crowley-Martin functional responses; see [44] and the references cited therein. The Crowley-Martin functional response can be seen as the generalization of the Holling II functional response, ratio-dependent functional response, and Beddition-DeAngelis functional response.…”

A Stochastic Holling-Type II Predator-Prey Model with Stage Structure and Refuge for Prey

“…There are three basic relationships among different species living in the same natural environment: (i) competitive relationship [1,2], for example, crops and weeds compete for fertilizer, sunlight, and other resources for their own growth; (ii) reciprocal relationship [3,4], such as rhinoceros birds feed on the parasites from rhinoceroses. For rhinoceroses, rhinoceros birds can get rid of the parasites and warn them when they are in danger; (iii) predation relationship [5,6], as the saying goes, "big fish eat small fish, small fish eat shrimps." Among the relationships of population interaction, the predator-prey system has been extensively studied.…”