Incorporating Prey Refuge Research Articles

Dynamical behavior of a discrete-time predator–prey system incorporating prey refuge and fear effect

This paper explores the effects of fear and refuge in a discrete-time two species predator–prey model. The fraction of prey that seeks refuge is assumed to be proportional to the amount of predators. The local stability of the corresponding fixed points is analyzed. The conditions of flip bifurcation and Neimark–Sacker bifurcation at the positive fixed point are described. The sufficient conditions for the positive fixed point to be globally asymptotically stable are derived in this article. Utilization of hybrid control strategy stabilizes the system’s chaotic nature. Finally, the paper supports the theoretical analysis with a numerical simulation that demonstrates the intricate nature of the system. It is observed that both fear effect and prey refuge can destabilize or stabilize the system depending on the value of handling time. Also, fear effect can restore stability in the system that exhibits chaos due to a high prey growth rate.

Dynamics of a stochastic prey–predator system with prey refuge, predation fear and its carry-over effects

This paper proposes and studies the dynamics of a Holling-type II predator–prey interaction system that incorporates the following three components: (1) a prey refuge; (2) predation fear and its carry-over effects; and (3) environmental noise in both prey and predator populations. The impacts of those three components are studied through both rigorous analysis and numerical simulations. Analytical results show that the introduction of prey refuge, predation fear, and its carry-over effects can generate Hopf bifurcation. It is found that increasing prey refuge and predation fear effect in a reasonable region can stabilize the system, while excessive refuge strength would lead to the extinction of predators. The theoretical results of the corresponding system with environmental noise include (1) sufficient conditions for the existence of a unique ergodic stationary distribution of the SDE system by constructing appropriate stochastic Lyapunov functions; (2) the explicit probability density function of the distribution by solving the Fokker–Planck equation; and (3) the extinction conditions of prey and/or predator species at an exponential rate in the long run. Our work shows that the proposed model, incorporating prey refuge, predation fear, carry-over effect, and environmental noise, exhibits rich and complex dynamic behaviors. Moreover, our results indicate that small environmental noise can save the prey and predator from extinction, while large environmental noise can drive the species to extinction. These interesting findings provide more perspectives on the protection and control of species in complex communities.

Effects of prey refuge with Holling type IV functional response dependent prey predator model

ABSTRACT In the present study, we have proposed two mathematical models for predator-prey interactions incorporating prey refuge having significant role in curbing the dynamical behaviour of both the system of prey-predator interactions from both the ecological and mathematical point of view. Here one introduces the prey refuge which depends on the encounters between prey and predator with Holling type IV functional response. Intra-specific competition among predators in searching food is considered in the second model as an additional effect to the first one. After the formulation of the models, the points of equilibria with its stability are systematically analyzed and the existence of bifurcation at the points of equilibrium has been duly carried out through their graphical representations with appropriate discussion in order to validate the applicability of both the system under consideration. It is found that the Hopf bifurcation of both the systems, which is of sub-critical type. The predation rate bears an important role in changing the stability of predator-free equilibrium to co-existence equilibrium. It is also observed that the biomass of prey density increases and predator density decreases with respect to gradual increment of the coefficient of prey refuge. Existence of paradox of enrichments are examined as well in both the model system.

Switching effect on a two prey–one predator system with strong Allee effect incorporating prey refuge

In an environment, the food chains are balanced by the prey–predator interactions. When a predator species is provided with more than one prey population, it avails the option of prey switching between prey species according to their availability. So, prey switching of predators mainly helps to increase the overall growth rate of a predator species. In this work, we have proposed a two prey–one predator system where the predator population adopts switching behavior between two prey species at the time of consumption. Both the prey population exhibit a strong Allee effect and the predator population is considered to be a generalist one. The proposed system is biologically well-defined as the system variables are positive and do not increase abruptly with time. The local stability analysis reveals that all the predator-free equilibria are saddle points whereas the prey-free equilibrium is always stable. The intrinsic growth rates of prey, the strong Allee parameters, and the prey refuge parameters are chosen to be the controlling parameters here. The numerical simulation reveals that in absence of one prey, the other prey refuge parameter can change the system dynamics by forming a stable or unstable limit cycle. Moreover, a situation of bi-stability, tri-stability, or even multi-stability of equilibrium points occurs in this system. As in presence of the switching effect, the predator chooses prey according to their abundance, so, increasing refuge in one prey population decreases the count of the second prey population. It is also observed that the count of predator population reaches a comparatively higher value even if they get one prey population at its fullest quantity and only a portion of other prey species. So, in the scarcity of one prey species, switching to the other prey is beneficial for the growth of the predator population.

Dynamical behaviour of a prey-predator system in a destructive environment incorporating prey refuge

Degradation of habitat is a direct outcome of anthropogenic activities, which includes urbanization, mining, the emission of industrial waste, and many others. Many living organisms experience severe surviving challenges as a result of habitat degradation. Here, we have studied the impact of habitat destruction caused by human activities on the dynamics of a prey-predator interaction with prey refuge. Our analyses reveal that a higher rate of habitat destruction than the habitat regeneration rate is always detrimental to the survival of predators. Predator species may still be threatened with extinction even if the rate of habitat degradation is slightly lower than the habitat recovery rate. So, in order to maintain biodiversity, we must appropriately step up our efforts to slow down the rate of habitat degradation as well as accelerate the habitat restoration. Further, our investigation suggests that in order to achieve cohabitation, we should effectively control the habitat deterioration caused by human activity, rather than artificially introducing or eliminating the hiding places of prey species.

Impact of Allee and fear effects in a fractional order prey-predator system incorporating prey refuge.

In this paper, we study the dynamic behaviors of a fractional order predator-prey system, in which the prey population has three effects: Allee effect, fear effect, and shelter effect. First, we prove in detail the positivity, existence, uniqueness, and boundedness of the solutions of the model from the perspective of mathematical analysis. Second, the stability of the system is considered by analyzing the stability of all equilibria and possible bifurcations of the system. It is proved that the system undergoes Hopf bifurcation with respect to four important parameters at the positive equilibrium point. Third, through stability analysis of the system, we find that: (i) as long as the initial density of the prey population is small enough, it will enter the attraction region of an extinction equilibrium point, making the system population at risk of extinction; (ii) we can eliminate the limit-cycle to make the system achieve stable coexistence by appropriately increasing the fear level or refuge rate, or reducing the prey natality or the order of fractional order systems; (iii) fractional order system is more stable than integer order systems, when the system has periodic solution, the two species can coexist stably by increasing the fear level or refuge rate appropriately. The threshold of fear level and refuge rate in fractional order systems is smaller than that in integer order systems. Finally, the rationality of the research results is verified by numerical simulation.

Stability and Boundedness Analysis of Lotka-Volterra Prey-Predator Model with Prey Refuge and Predator Cannibalism

Abstract In this paper, a prey-predator system incorporating prey refuge and predator cannibalism is studied. The stability and ultimate boundedness of the analyzed state parameters (y_1,y_2) defining the system are obtained using the Lyapunov’s second or direct method. We construct a suitable complete Lyapunov function for the nonlinear system and demonstrate its efficacy. The method is built upon applying various theoretical Lyapunov functions. By constructing a Lyapunov function which possesses a functional relationship to the the original model system, we give sufficient conditions which ensure the stability and ultimate boundedness of the state parameters describing the nonlinear prey-predator system. We give a numeric example to illustrate the result obtained. JEL classification numbers: 34C27, 34D23, 34C25. Keywords: Prey-predator system, Prey refuge, Predator cannibalism, Stability, Boundedness, Lyapunov’s method.

Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives

Most of the preys are well aware of sensing predation risk. Consequently, to escape from predators they usually adopt several defense mechanisms, specially refuge themselves to become invulnerable. In view of this, a mathematical model has been formulated incorporating prey refuge, where it is assumed that prey refuge is a function of predators availability in the system. It is shown that the model system is well-posed. It has been found that the hiding level and consumption rate of predators have a suitable interrelation between them. Both the parameters act as Hopf bifurcation parameters, but they play opposite role in case of stabilization of the system dynamics. Also, hiding level plays crucial role in maintaining the mean density of both the populations. Furthermore, as hiding behaviour of prey is not instantaneous, so a time delay, namely hiding delay has been introduced to make the model system more realistic and it is observed that the delay parameter destabilizes the system. Modelling approach through fractional calculus has been further deployed to study how the process of forgetting life history influences the dynamical intricacy of the population level dynamics. All the analytical findings have been testified by proper numerical performances.

Biological control of infection pervasive via pest: a study of prey–predator model incorporating prey refuge under fuzzy impreciseness

ABSTRACT The prey–predator systems exist in almost all ecological systems and the refuge stage is highly concerned with such ecological systems in which the prey goes out of reach of the attacking region of the predator. In this paper, dynamical behaviour of predator and prey species with Holling type III response function is investigated. We add refuge in prey species and also additional predators are introduced into the system. Firstly, the crisp model is formulated with these effects and the boundedness, existence, conditions of local and global stability, local bifurcation and Hopf bifurcation are discussed. The crisp model is transformed into the fuzzy model to incorporate the unpredictable behaviour and using the Root Mean Square (RMS) defuzzification method, the fuzzy model is reduced to the defuzzified model in order to avoid the complexity of the analysis of the solution. Further, numerical simulations are performed with the help of MATLAB, MATCONT package and we examine solution curves and phase portraits in two different cases, in the presence and in the absence of additional predators along with the Hopf bifurcation scenarios. Finally, the sensitivities of the controlling parameters are plotted graphically and presented their significances in controlling the infection of the prey.

Effect of Fear in Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response Incorporating Prey Refuge

In the present paper, we study the effect of antipredator behavior due to fear of predation on a modified Leslie-Gower predator-prey model incorporating prey refuge which predation rate of predators follows Beddington-DeAngelis functional response. The biological justification of the model is demonstrated through non-negativity, boundedness, and permanence. Next, we perform the analysis of equilibrium and local stability. We obtain four equilibrium points where two points are locally asymptotically stable and other points are unstable. Besides, we show the effect of the fear in the model and obtain a conclusion that the increased rate of fear can decrease the density of both populations, and prey populations become extinct. Meanwhile, for the case with a constant rate of fear, the prey refuge helpful to the existence of both populations. However, for the case with the fear effect is large, prey refuge cannot cause the extinction of predators. Several numerical simulations are performed to support our analytical results.

Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases. In the end, several numerical solutions are presented to check the analytical results.

Effect of Prey Refuge and Harvesting on Dynamics of Eco-epidemiological Model with Holling Type III

In this research, we formulate and analyze an eco-epidemiology model of the modified Leslie-Gower model with Holling type III by incorporating prey refuge and harvesting. In the model, we find at most six equilibrium where three equilibrium points are unstable and three equilibrium points are locally asymptotically stable. Furthermore, we find an interesting phenomenon, namely our model undergoes Hopf bifurcation at the interior equilibrium point by selecting refuge as the bifurcation parameter. Moreover, we also conclude that the stability of all populations occurs faster when the harvesting rate increases. In the end, several numerical solutions are presented to check the analytical results.

Analysis of a tritrophic food chain model with fear effect incorporating prey refuge

In this work, we have introduced a tritrophic food-chain model where consumer hunt for prey with Holling type-III functional response. The birth rate of the prey population has been reduced due to the fear of predation, i.e., a fear effect is considered in the prey population. Moreover, a fraction of the prey is available to the consumer for consumption and this has been done by incorporation of prey refuge term. The predation between consumer and predator follows Beddington-DeAngelis response. Boundedness and positivity of the system prove that the proposed model is well-posed. Also, there are some parametric restrictions under which the system is permanent. Routh-Hurwitz criterion shows the local stability conditions of the equilibrium points and on the other hand Lyapunov LaSalle theorem guarantees that the locally stable equilibrium points are globally stable. Also, Matlab validates the analytical results with the help of diagrams. The occurrence of transcritical bifurcations have been shown and conditions for the existence of a limit cycle in the system through Hopf bifurcation also have been stated. Both the analytical and numerical results suggest that a certain amount of fear can make the system steady. It is also noted that the prey refuge has both stabilizing and destabilizing effect on the system.

Pattern dynamics of a reaction-diffusion predator-prey system with both refuge and harvesting

We are concerned with a reaction-diffusion predator–prey model under homogeneous Neumann boundary condition incorporating prey refuge (proportion of both the species) and harvesting of prey species in this contribution. Criteria for asymptotic stability (local and global) and bifurcation of the subsequent temporal model system are thoroughly analyzed around the unique positive interior equilibrium point. For partial differential equation (PDE), the conditions of diffusion-driven instability and the Turing bifurcation region in two-parameter space are investigated. The results around the unique interior feasible equilibrium point specify that the effect of refuge and harvesting cooperation is an important part of the control of spatial pattern formation of the species. A series of computer simulations reveal that the typical dynamics of population density variation are the formation of isolated groups within the Turing space, that is, spots, stripe-spot mixtures, labyrinthine, holes, stripe-hole mixtures and stripes replication. Finally, we discuss spatiotemporal dynamics of the system for a number of different momentous parameters via numerical simulations.

Dynamical analysis of a prey-predator model incorporating a prey refuge with variable carrying capacity

A prey-predator model incorporating prey refuge with variable carrying capacity and Holling type-II functional response is proposed and analyzed. The model includes a case of increasing carrying capacity as well as a decreasing carrying capacity case. Sufficient conditions are derived to ensure the existence and local stability of the equilibrium points of the proposed model. Moreover, the occurrence of transcritical bifurcation as well as Hopf bifurcation are investigated. The effect of some model parameter related to the prey refuge and the variable carrying capacity on the prey-predator dynamics has been examined. Numerical simulations are presented to demonstrate the theoretical results and to illustrate the effect of these parameters on the model dynamics. Moreover, a comparison with the constant carrying case has been presented.

Dynamical study of a prey–predator model incorporating nonlinear prey refuge and additive Allee effect acting on prey species

In this study, we have analyzed a mathematical model on predator–prey interactions incorporating prey refuge and additive Allee effect on the prey species. The various dynamical behaviors of the system have analyzed, considering the prey refuge is proportional to both the prey and predator species with Beddington–DeAngelis functional response. None, single, or two coexistence equilibria can exist at the first quadrant of the phase space considering strong additive Allee effect in the system. The permanence, local stability, saddle-node bifurcation, existence of a stable limit cycle and Hopf bifurcation are examined under some parametric conditions. We have also calculated the first Lyapunov number to define the nature of Hopf bifurcating periodic solution. Moreover, it has established a parameter subset at which the dynamical system may have a cusp point of co dimension 2 (Bogdanov–Takens bifurcation). Finally, we have executed an adequate numerical simulation to authenticate our analytical findings.

Time-delayed predator–prey interaction with the benefit of antipredation response in presence of refuge

Abstract A field experiment on terrestrial vertebrates observes that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect effect of predator’s fear (felt by prey) through chemical and/or vocal cues may also reduce the reproduction of prey and change their life history. In this work, we have described a predator–prey model with Holling type II functional response incorporating prey refuge. Irrespective of being considering either a constant number of prey being refuged or a proportion of the prey population being refuged, a different growth rate and different carrying capacity for the prey population in the refuge area are considered. The total prey population is divided into two subclasses: (i) prey x in the refuge area and (ii) prey y in the predatory area. We have taken the migration of the prey population from refuge area to predatory area. Also, we have considered a benefit from the antipredation response of the prey population y in presence of cost of fear. Feasible equilibrium points of the proposed system are derived, and the dynamical behavior of the system around equilibria is investigated. Birth rate of prey in predatory region has been regarded as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the interior equilibrium point. Moreover, the conditions for occurrence of transcritical bifurcations have been determined. Further, we have incorporated discrete-type gestational delay on the system to make it more realistic. The dynamical behavior of the delayed system is analyzed. Finally, some numerical simulations are given to verify the analytical results.

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Disease control prey–predator model incorporating prey refuge under fuzzy uncertainty

Ecological system depends on prey predator interaction. Sometimes diseases are spread among prey or predator or both species. In this paper, a disease control prey–predator model with Holling type-II functional response incorporating refuge in the susceptible prey is proposed. The impact of additional food for predator species in the system is investigated. Using some assumptions, the crisp model is formulated and it is converted into fuzzy model and defuzzified models by using basic defuzzification method. In the theoretical section, the condition of boundedness, existence of equilibrium points, global stability of interior equilibrium point and Hopf bifurcation on a refuge parameter are investigated. Theoretical results are verified in our numerical simulation section. Using MATLAB package we examined the behaviour of the species in the presence and absence of additional food in both crisp and fuzzy environments. The phase trajectories for different initial conditions in both environments and variations in population of the species are presented. Using MATCONT package, we present the bifurcation scenarios when the additional food parameter and refuge parameter vary. We compute the existence of Hopf point (H) and branch point (BP) in the model for suitable supply of additional food and intensity of refuge parameter. Finally the sensitivities of the parameters are plotted graphically. This theoretical and numerical study may be useful to control the infectious diseases in real world ecological systems.

A delayed fractional order food chain model with fear effect and prey refuge

A delayed fractional-order prey–predator system with fear (felt by prey) effect of predator on prey population incorporating prey refuge has been proposed. We consider a three species food chain system with Holling type I functional response for the predator population including prey refuge. The existence and uniqueness of the system is studied along with non-negativity and boundedness of the solutions of proposed system. Next, local stability of the equilibria has been studied for both delayed and non-delayed systems. We have also established that the non delayed system is globally stable under some parametric restrictions. Finally we have discussed the Hopf bifurcation due to time delay and other parameters both theoretically and numerically by the help of MATLAB and MAPLE.