Interactive effects of prey refuge and additional food for predator in a diffusive predator-prey system

Additional food for predators has been considered as one of the best established techniques in integrated pest management and biological conservation programs. In natural systems, there are several other factors, e.g., prey refuge, affect the success of pest control. In this paper, we analyze a predator-prey system with prey refuge and additional food for predator apart from the focal prey in the presence of diffusion. Our main aim is to study the interactive effects of prey refuge and additional food on the system dynamics and especially on the controllability of prey (pest). Different types of Turing patterns such as stripes, spots, holes, and mixtures of them are obtained. It is found that the supply of additional food to the predator is unable to control the prey (pest) population when prey refuge is high. Moreover, when both prey refuge and additional food are low, spatial distribution of prey becomes complex and once again prey control becomes difficult. However, the joint effect of reduction in prey refuge and the presence of appropriate amount of additional food can control prey (pest) population from the system.     

Stability and bifurcation of a stage-structured predator–prey model with both discrete and distributed delays

This paper concerns with a new delayed predator–prey model with stage structure on prey, in which the immature prey and the mature prey are preyed by predator and the delay is the length of the immature stage. Mathematical analysis of the model equations is given with regard to invariance of non-negativity, boundedness of solutions, permanence and global stability and nature of equilibria. Our work shows that the stage structure on the prey is one of the important factors that affect the extinction of the predator, and the predation on immature prey is a cause of periodic oscillation of population and can make the behaviors of the system more complex. The predation on the immature and mature prey brings both positive and negative effects on the permanence of the predator, if ignore the predation on immature prey in the system, the stage-structure on prey brings only negative effect on the permanence of the predator.     

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.     

Uniqueness and stability of positive steady state solutions for a ratio-dependent predator–prey system with a crowding term in the prey equation

This paper deals with a ratio-dependent predator–prey system with a crowding term in the prey equation, where it is assumed that the coefficient of the functional response is less than the coefficient of the intrinsic growth rates of the prey species. We demonstrate some special behaviors of solutions to the system which the coexistence states of two species can be obtained when the crowding region in the prey equation only is designed suitably. Furthermore, we demonstrate that under some conditions, the positive steady state solution of the predator–prey system with a crowding term in the prey equation is unique and stable. Our result is different from those ones of the predator–prey systems without the crowding terms.     

Modelling a predator–prey system with infected prey in polluted environment

A predator–prey model with logistic growth in prey is modified by introducing an SIS parasite infection in the prey. We have studied the combined effect of environmental toxicant and disease on prey–predator system. It is assumed in this paper that the environmental toxicant affects both prey and predator population and the infected prey is assumed to be more vulnerable to the toxicant and predation compared to the sound prey individuals. Thresholds are identified which determine when system persists and disease remains endemic.     

Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.     

The impact of Allee effect on a predator–prey system with Holling type II functional response

In this paper, the Allee effect is incorporated into a predator–prey model with Holling type II functional response. Compared with the predator–prey model without Allee effect, we find that the Allee effect of prey species increases the extinction risk of both predators and prey. When the handling time of predators is relatively short and the Allee effect of prey species becomes strong, both predators and prey may become extinct. Moreover, it is shown that the model with Allee effect undergoes the Hopf bifurcation and heteroclinic bifurcation. The Allee effect of prey species can lead to unstable periodical oscillation. It is also found that the positive equilibrium of the model could change from stable to unstable, and then to stable when the strength of Allee effect or the handling time of predators increases continuously from zero, that is, the model admits stability switches as a parameter changes. When the Allee effect of prey species becomes strong, longer handling time of predators may stabilize the coexistent steady state.     

A bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting

In this paper, we propose a bioeconomic differential algebraic predator–prey model with Holling type II functional response and nonlinear prey harvesting. As the nonlinear prey harvesting is introduced, the proposed model displays a complex dynamics in the predator–prey plane. Taking into account of the economic factor, our predator–prey system is established by bioeconomic differential algebraic equations. The effect of economic profit on the proposed model is analyzed by viewing it as a bifurcation parameter. By jointly using the normal form of differential algebraic models and the bifurcation theory, the stability and bifurcations (singularity induced bifurcation, Hopf bifurcation) are discussed. These results obtained here reveal richer dynamics of the bioeconomic differential algebraic predator–prey model with nonlinear prey harvesting, and suggest a guidance for harvesting in the practical word. Finally, numerical simulations are given to demonstrate the results.     

Effect of emergent carrying capacity in an eco‐epidemiological system

The paper explores an eco‐epidemiological model of a predator–prey type, where the prey population is subject to infection. The model is basically a combination of S‐I type model and a Rosenzweig–MacArthur predator–prey model. The novelty of this contribution is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We explicitly separate the competition between non‐infected and infected individuals. This emergent carrying capacity is markedly different to the explicit carrying capacities that have been considered in many eco‐epidemiological models. We observed that different intra‐class and inter‐class competition can facilitate the coexistence of susceptible prey‐infected prey–predator, which is impossible for the case of the explicit carrying capacity model. We also show that these findings are closely associated with bi‐stability. The present system undergoes bi‐stability in two different scenarios: (a) bi‐stability between the planner equilibria where susceptible prey co‐exists with predator or infected prey and (b) bi‐stability between co‐existence equilibrium and the planner equilibrium where susceptible prey coexists with infected prey; have been discussed. The conditions for which the system is to be permanent and the global stability of the system around disease‐free equilibrium are worked out. Copyright © 2015 John Wiley & Sons, Ltd.     

An eco‐epidemiological predator–prey model where predators distinguish between susceptible and infected prey

A predator–prey model with disease amongst the prey and ratio‐dependent functional response for both infected and susceptible prey is proposed and its features analysed. This work is based on previous mathematical models to analyse the important ecosystem of the Salton Sea in Southern California and New Mexico where birds (particularly pelicans) prey on fish (particularly tilapia). The dynamics of the system around each of the ecologically meaningful equilibria are presented. Natural disease control is considered before studying the impact of the disease in the absence of predators and the interaction of predators and healthy prey and the disease effects on predators in the absence of healthy prey. Our theoretical results are confirmed by numerical simulation. Copyright © 2016 John Wiley & Sons, Ltd.     

Bifurcations in a Leslie–Gower model with constant and proportional prey refuge at high and low density

Assuming that the prey refuge is proportional to the prey density if its population size is below a critical threshold, or constant if its size is above the threshold, this paper proposes, and qualitatively analyzes, a Leslie–Gower predator–prey model assuming alternative feeding and harvesting in predators, and a Holling II function as the predator functional response. From the results of the mathematical analysis to the predator–prey models with proportional or constant prey refuge, the proposed model retains the same bifurcation cases obtained for each model analyzed. However, appropriate alterations of the parameters representing the critical threshold of prey population size and harvest in predators allows the formation of at least one limit cycle, stable or unstable, that lives in both vector fields of the proposed model.     

Predator-prey dynamics with square root functional responses

A predator-prey model is considered in which a modified Lotka-Volterra interaction term is used as the functional response of the predator to the prey. The interaction term is proportional to the square root of the prey population, which appropriately models systems in which the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd. Because of the square root term, the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

HARVESTING A TWO-PATCH PREDATOR-PREY METAPOPULATION

ABSTRACT. A mathematical model for a two-patch predator-prey metapoplation is developed as a generalization of single-species metapopulation harvesting theory. We find optimal harvesting strategies using dynamic programming and La-grange multipliers. If predator economic efficiency is relatively high, then we should protect a relative source prey subpopulation in two different ways: directly, with a higher escapement of the relative source prey subpopulation, and indirectly, with a lower escapement of the predator living in the same patch as the relative source prey subpopulation. Numerical examples show that if the growth of the predator is relatively low and there is no difference between prey and predator prices, then it may be optimal to harvest the predator to extinction. While, if the predator is more valuable compared to the prey, then it may be optimal to leave the relative exporter prey subpopulation unharvested. We also discuss how a ‘negative’ harvest might be optimal. A negative harvest might be considered a seeding strategy.     

Classical predator–prey system with infection of prey population—a mathematical model

The present paper deals with the problem of a classical predator–prey system with infection of prey population. A classical predator–prey system is split into three groups, namely susceptible prey, infected prey and predator. The relative removal rate of the susceptible prey due to infection is worked out. We observe the dynamical behaviour of this system around each of the equilibria and point out the exchange of stability. It is shown that local asymptotic stability of the system around the positive interior equilibrium ensures its global asymptotic stability. We prove that there is always a Hopf bifurcation for increasing transmission rate. To substantiate the analytical findings, numerical experiments have been carried out for hypothetical set of parameter values. Our analysis shows that there is a threshold level of infection below which all the three species will persist and above which the disease will be epidemic. Copyright © 2003 John Wiley & Sons, Ltd.     

随机配对的食饵-捕食者模型

建立了食饵具有性别结构的食饵-捕食者模型,其中食饵两性的配对是随机的.结论表明:没有偏食的捕食不能改变食饵的性比,偏食可以改变食饵的性比;若食饵两性的出生率相等,任意的偏食都可以消除周期振荡;若食饵两性的出生率不相等,较大的偏食总能消除周期振荡,周期振荡的消除与否由偏食系数和食饵两性的出生率之比共同决定.     

Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone

This paper is concerned with the stationary problem of a prey-predator cross-diffusion system with a protection zone for the prey. We discuss the existence and non-existence of coexistence states of the two species by using the bifurcation theory. As a result, it is shown that the cross-diffusion for the prey has beneficial effects on the survival of the prey when the intrinsic growth rate of the predator is positive. We also study the asymptotic behavior of positive stationary solutions as the cross-diffusion coefficient of the prey tends to infinity.     

A numerical study of the formation of spatial patterns in twospotted spider mites

The aim of this paper is to study the formation of spatial patterns in a predator–prey system with Tetranychus urticae as prey and Phytoseiulus persimilis as predator. Logistic Lotka–Volterra predator–prey equations are solved numerically with two different response functions, two initial conditions and one data set. The spatial patterns are generated by introducing diffusion-driven instability in the predator–prey system. Among all parameters involved in predator–prey equations, only the predator interference parameter is varied to generate diffusion-driven instability leading to spatial patterns of population density. Spatial patterns are further generated with the inclusion of prey-taxis in the predator–prey system. Routh–Hurwitz’s conditions for stability are used to create instability with prey-taxis in the system. It is shown that it is possible to generate spatial patterns with zero flux boundary conditions even in a smaller domain with a suitable value of the predator interference parameter or prey-taxis.     

A predator–prey model with disease in the prey and two impulses for integrated pest management

In this paper, a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. In the first part of the main results, the sufficient condition for the global stability of the susceptible pest-eradication periodic solution is obtained, which means if the release amount of infective prey and predator satisfy the condition, then the pest will be controlled. The sufficient condition for the permanence of the system is also obtained subsequently, which means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. At last, we interpret our mathematical results.     

Coexistence in seasonally varying predator–prey systems with Allee effect

A general seasonally-varying predator–prey model with Allee effect in the prey growth is investigated. The analysis is performed only on the basis of some properties determining the shape of the prey growth rate and the functional responses. General conditions for coexistence are determined, both in the case of weak and strong Allee effect. Finally, a modified Leslie–Gower predator–prey model with Allee effect is investigated. Numerical results illustrate the qualitative behaviors of the system, in particular the presence of periodic orbits.