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The aim of the paper is to study the impact of anti-predator behavior caused by dread of predator species in a prey predator system with Holling III type functional response and prey shelters. Firstly, we analyze the dynamic behavior of the system, including the stability of the system and demonstrating the occurrence of Hopf bifurcation around the positive equilibrium point and the existence of limit cycle emerging through Hopf bifurcation. Secondly, through the study of the effect of fear and refuge, we discover that the increase of fear level can improve the stability of the system by eliminating periodic solutions and decrease the populations of predator species at the coexist equilibrium, but not cause the extinction of the predators, and prey refuge also plays very vital role in the persistence of the predators. Finally, the rationality of the results is verified by numerical simulation.

We perform a detailed analysis of the behaviour of a non-autonomous prey–predator model where age-based growth with age discriminatory harvesting in prey and predator’s reliance upon alternative food in the absence of that particular prey are considered. We begin by deriving certain sufficient conditions for permanence and positive invariance and then proceed to construct a Lyapunov function to derive some constraints for global attractivity. With the help of continuation theorem, we arrive at the best fit criterion to prove the occurrence of a positive periodic solution. Moreover, using Arzela–Ascoli theorem, we formulate a proof for a unique positive solution to be almost periodic, and we carry out numerical simulation to verify the analytical findings. With the aid of graphs and tables, we show the nature of the prey–predator system in response to the alternative food and delays.

A prey–predator model with Ivlev-type functional response and the fear effect on prey species by the predator have been considered for the first time in a crisp as well as fuzzy environment. The effects of fear have been investigated on the stability of the system. Granular function derivative concept has been used to do fuzzy mathematics. For the first time, proper model analysis, positivity, bounds and uniform persistence are studied for our proposed model in fuzzy environment. The conditions of stability of all co-existing equilibrium points and Hopf bifurcation analysis have also been studied in fuzzy environment. Analytical results have been justified by numerical simulation with proper table and graphical presentation in crisp and fuzzy environment both.

Prey-predator models with refuge effect have great importance in the context of ecology. Constant refuge and refuge proportional to prey are the most popular concepts of refuge in the existing literature. Now, there are new different types of refuge concepts attracting researchers. This study considers a refuge concept proportional to the predator due to the fear induced by predators. When predators increase, fears also increase and that is why prey refuges also increase. Here, we examine the influence of prey refuge proportional to predator effect in a discrete prey-predator interaction with the Holling type II functional response model. Is this refuge stabilizing or destabilizing the system? That is the central question of this study. The existence and stability of fixed points, Period-Doubling Bifurcation, Neimark–Sacker Bifurcation, the influence of prey refuge, and chaos are analyzed. This work provides the bifurcation diagrams and Lyapunov exponents to analyze the refuge parameter of the model. The proposed discrete model indicates rich dynamics as the effect of prey refuge through numerical simulations.

In this paper, we propose and investigate a prey–predator model with Holling type II response function incorporating Allee and fear effect in the prey. First of all, we obtain all possible equilibria of the model and discuss their stability by analyzing the eigenvalues of Jacobian matrix around the equilibria. Secondly, it can be observed that the model undergoes Hopf bifurcation at the positive equilibrium by taking the level of fear as bifurcation parameter. Moreover, through the analysis of Allee and fear effect, we find that: (i) the fear effect can enhance the stability of the positive equilibrium of the system by excluding periodic solutions; (ii) increasing the level of fear and Allee can reduce the final number of predators; (iii) the Allee effect also has important influence on the permanence of the predator. Finally, numerical simulations are provided to check the validity of the theoretical results.

A modified version of the so called Holling-Tanner prey-predator models with prey dependent functional response is introduced. We improved some new results on Holling-Tanner model from Lotka-Volterra model on real ecological systems and studied the stability of this model in the deterministic and stochastic environments. The study was focused on three types of stability, namely, stable node, spiral node, and center. The numerical schemes are employed to get the approximated solutions of the differential equations. We have used Euler scheme to solve the deterministic prey-predator model and we used Euler-Maruyama scheme to solve stochastic prey-predator model.

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

We explore the dynamics of a time-variant prey–predator model of two species of fishes, namely Tenualosa ilisha (prey) and Macrognathus pancalus (predator). The prey exhibits anti-predatory behavior due to fear of predation. The prey equation is also equipped with age-based growth and harvestation terms to investigate the significance of maturity and harvest delay. The prey moves from saline water to freshwater for spawning, during which the predator feeds on an alternative food source. We explore the impact of this food source on the survival of prey. Starting with the sufficient conditions for positivity and boundedness of the solutions to the model, we find conditions for the existence of a stable global solution, periodic and almost periodic solutions. We find the range of existence for prey and predator populations and observe the interdependence between these values. According to our findings, just the fear of predation is enough to drive the prey population low. Even when the prey happens to be in an advantageous scenario, fear can still bring its population down, unless there are no or very few predators.

This paper studies the periodic trajectories of a novel age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion. To begin with, the system is turned into an abstract non-densely defined Cauchy problem, and a time-lag effect in their interaction is investigated. Next, we acquire that this system appears a periodic orbit near the positive steady state by employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, which is also the main result of this article. Finally, in order to illustrate our theoretical analysis more vividly, we make some numerical simulations and give some discussions.

In the present paper we have investigated the impact of predator harvesting in a two-dimensional prey–predator model with Holling type III functional response. The main objective of this paper is t...

This paper investigates the stability of bifurcating solutions for a prey-predator model with population flux by attractive transition. Applying spectral analysis and the principle of exchange of stability, we obtain that the bifurcating solutions are stable/unstable under some certain conditions.

In this paper, an eco-epidemiological model with media coverage effects is established and studied. An -type of disease in predator is considered. All the properties of the solution of the proposed model are discussed. An application to the stability theory was carried out to investigate the local as well as global stability of the system. The persistence conditions of the model are determined. The occurrence of local bifurcation in the model is studied. Further investigation of the global dynamics of the model is achieved through using a numerical simulation.

In this paper, a two-component generic prey-predator system incorporated with habitat complexity in predator functional response, and with constant time delay in predator gestation is considered. Although the role of time delay on the system dynamics is widely studied in the literature, only a few researchers have addressed the effect of habitat complexity in the prey-predator type interactions. In the first part of the paper the equilibria and stability analysis of the mathematical model is mentioned. In the second part, particular attention is paid on the numerical bifurcation analysis of the prey and predator densities based on two system parameters:(i) the strength of homogeneous habitat complexity and (ii) predator attack rate with and without time delay. It is found that dynamics with time delay in predator gestation are found to be much richer compared to that without time delay. The system stability may change from stable to unstable through a Hopf bifurcation and the solution branches emanating from these Hopf points are usually stable and supercritical. However, delay driven system may lead unstable orbits arising from Hopf bifurcations. It is also found that increasing the strength of habitat complexity may lead the stability change from unstable to stable.

This paper studies a reaction–advection–diffusion prey–predator system in one spatial dimension. Adapting the Lotka–Volterra-type functional response, we prove the global existence and boundedness ...

The importance of fear, refuge and additional food is being increasingly recognized in recent studies, but their combined effects need to be explored. In this paper, we investigate the joint effects of these three ecologically important factors in a prey–predator system with Crowly–Martin type functional response. We find that the fear of predator significantly affects the densities of prey and predator populations whereas the presence of prey refuge and additional food for predator are recognized to have potential impacts to sustain prey and predator in the habitat, respectively. The fear of predator induces limit cycle oscillations while an oscillatory system becomes stable on increasing the refuge. The system first undergoes a supercritical Hopf-bifurcation and then a subcritical Hopf-bifurcation on increasing either the growth rate of prey or growth rate of predator due to additional food. Increasing the quality/quantity of additional food after a certain value causes extinction of prey species and rapid incline in the predator population. An extension is made in the model by considering the seasonal variations in the cost of fear of predator, prey refuge and growth rate of predator due to additional food. The nonautonomous model is shown to exhibit globally attractive positive periodic solution. Moreover, complex dynamics such as higher periodic solutions and bursting patterns are observed due to seasonal variations in the rate parameters.

The production and quality of cocoa beans remain key issues for the economy of Cote dIvoire. The fairly humid Ivorian climate promotes increased mold production, leading therefore to a considerable negative impact on beans quality. In this work, a prey predator model was proposed to simulate the molds growth on cocoa beans. Our model (MPM), based on that of Leslie Gower, was identified using a multi-objective genetic algorithm (MOGA). The developed model allows to stimulate the relationship between parameters (i.e. sugar content (Su) and molds (M) with a good accuracy (R=0.9963 and 0.8382 respectively for molds growth and sugar content).

Parameters involve in various ecological models are mostly uncertain due to ever changing characteristic of nature. This article, deals with a basic predator–prey model where ecological parameters are considered as parametric-functional nature of interval numbers. The dynamical behaviors of the model system have been discussed in the imprecise environment. Moreover, the concept of maximum sustainable yield (MSY) has been incorporated in the model system under imprecise environment, and MSY policies under different harvesting scenarios have been discussed. Finally all the analytical findings are testified through extensive numerical simulations and the article ended with a conclusion.

The problem of finite-time stabilization for a Leslie-Gower prey – predator system through a bounded control input is solved. We use Korobov’s controllability function. The trajectory of the resulting motion is ensured for fulfilling a physical restriction that prey and predator cannot achieve negative values. For this purpose, a certain ellipse depending on given data and the equilibrium point of the considered system is constructed. Simulation results show the effectiveness of the proposed control methodology.

In this paper, the prey-predator model of five compartments is constructed with treatment given to infected prey and infected predator. We took predation incidence rates as functional response type II, and disease transmission incidence rates follow simple kinetic mass action function. The positivity, boundedness, and existence of the solution of the model are established and checked. Equilibrium points of the models are identified, and local stability analyses of trivial equilibrium, axial equilibrium, and disease-free equilibrium points are performed with the method of variation matrix and the Routh-Hurwitz criterion. It is found that the trivial equilibrium point E o is always unstable, and axial equilibrium point E A is locally asymptotically stable if β k − t 1 + d 2 < 0 , q p 1 k − d 3 s + k < 0 and q p 3 k − t 2 + d 4 s + k < 0 conditions hold true. Global stability analysis of an endemic equilibrium point of the model has been proven by considering the appropriate Lyapunov function. The basic reproduction number of infected prey and infected predators are obtained as R 01 = q p 1 − d 3 2 k β d 3 s 2 / q p 1 − d 3 q p 1 − d 3 2 k s t 1 + d 2 + r s q p 2 k q p 1 − k d 3 − d 3 s and R 02 = q p 1 − d 3 q p 3 d 3 k + α r s q k q p 1 − k d 3 − d 3 s / q p 1 − d 3 2 t 2 + d 4 k , respectively. If the basic reproduction number is greater than one, then the disease will persist in the prey-predator system. If the basic reproduction number is one, then the disease is stable, and if the basic reproduction number is less than one, then the disease dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.

In this paper, a stochastic prey-predator model is investigated and analyzed, which possesses foraging arena scheme in polluted environments. Sufficient conditions are established for the extinction and persistence in the mean. These conditions provide a threshold that determines the persistence in the mean and extinction of species. Furthermore, it is also shown that the stochastic system has a periodic solution under appropriate conditions. Finally, several numerical examples are carried on to demonstrate the analytical results.

In ectothermic predator-prey relationships, evasion of predation by prey depends on physiological and behavioural responses relating to the thermal biology of both predator and prey. On Japan's Izu Islands, we investigated a prey lizard's physiological and thermal responses to the presence of a snake predator over geologic time in addition to recent climatic warming. Foraging lizard body temperatures increased by 1.3 °C from 1981 to 2019 overall, yet were 2.9 °C warmer on snake islands relative to snake-free islands. We also detected snake predator-induced selection on hind leg length, which in turn is a major determinant for sprint speed only in lizard populations exposed to predation by snakes. Accordingly, we found that warmer prey body temperatures result in faster sprint speeds by the prey at temperatures suboptimal for the snake predator, and therefore contribute to escaping predation. Given recent climatic change, further warming could irrevocably alter this and other ectothermic predator-prey relationships.

A mathematical model for predator prey system with immigrant prey previously proposed is solved analytically to get an approximate analytical solution for local prey population density, immigrant prey population density and the predator population density. The analytical solution derived is compared with the numerical solution obtained using MATLAB and is found to make a good fit. The derived results have been illustrated with their graphs for certain parametric values.

Additional food provided prey-predator systems have become a significant and important area of study for both theoretical and experimental ecologists. This is mainly because provision of additional food to the predator in the prey-predator systems has proven to facilitate wildlife conservation as well as reduction of pesticides in agriculture. Further, the mathematical modeling and analysis of these systems provide the eco-manager with various strategies that can be implemented on field to achieve the desired objectives. The outcomes of many theoretical and mathematical studies of such additional food systems have shown that the quality and quantity of additional food play a crucial role in driving the system to the desired state. However, one of the limitations of these studies is that they are asymptotic in nature, where the desired state is reached eventually with time. To overcome these limitations, we present a time optimal control study for an additional food provided prey-predator system involving inhibitory effect with quantity of additional food as the control parameter with the objective of reaching the desired state in finite (minimum) time. The results show that the optimal solution is a bang-bang control with a possibility of multiple switches. Numerical examples illustrate the theoretical findings. These results can be applied to both biological conservation and pest eradication.

Fear of predation plays an important role in the growth of a prey species in a prey-predator system. In this work, a two-species model is formulated where the prey species move in a herd to protect themselves and so it acts as a defense strategy. The birth rate of the prey here is affected due to fear of being attacked by predators and so, is considered as a decreasing function. Moreover, there is another fear term in the death rate of the prey population to emphasize the fact that the prey may die out of fear of predator too. But, in this model, the function characterizing the fear effect in the death of prey is assumed in such a way that it is increased only up to a certain level. The results show that the system performs oscillating behavior when the fear coefficient implemented in the birth of prey is considered in a small amount but it changes its dynamics through Hopf bifurcation and becomes stable for a higher value of the coefficient. Regulating the fear terms ultimately makes an impact on the growth of the predator population as the predator is taken as a specialist predator here. The increasing value of the fear terms (either implemented in birth or death of prey) decrease the count of the predator population with time. Also, the fear implemented in the birth rate of prey makes a higher impact on the growth of the predator population than in the case of the fear-induced death rate.

In this paper, we consider a prey-predator model with a reserve region of predator where generalist predator cannot enter. Based on the intake capacity of food and other factors, we introduce the predator population which consumes the prey population with Holling type-II functional response; and generalist predator population consumes the predator population with Beddington-DeAngelis functional response. The density-dependent mortality rate for prey and generalist predator are considered. The equilibria of proposed system are determined. Local stability for the system are discussed. The environmental carrying capacity is considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighbourhood at an interior equilibrium point. Here the fishing effort is used as a control parameter to harvest the generalist predator population of the system. With the help of this control parameter, a dynamic framework is developed to investigate the optimal utilization of resources, sustainability properties of the stock and the resource rent. Finally, we present a numerical simulation to verify the analytical results, and the system is analyzed through graphical illustrations. The main findings with future research directions are described at last.

Employing the Lotka -Voltera (1926) prey-predator model equation, the system is presented with harvesting effort for both species prey and predator. We analyze the stability of the system of ordinary differential equation after calculating the Eigen values of the system. We include the harvesting term for both species in the model equation, and observe the dynamic analysis of prey-predator populations by including the harvesting efforts on the model equation. We also analyze the population dynamic of the system by varying the harvesting efforts on the system. The model equation are solved numerically by applying Runge - Kutta fourth order method.

This paper is designed based on the combined bioeconomic harvesting of Holling-Tanner prey-predator competition model with Beddington-DeAngelis functional response with two different delays. The si...

Dynamical characteristics of a stochastically perturbed system are drastically different from those of a deterministic system. This study deals with the dynamics of a stochastically perturbed time-variant prey–predator model of aquatic animals hilsa (prey) and eel (predator). It is a well-known fact that overfishing and indiscriminate fishing of juvenile hilsa have led to a drastic fall in its population. To save the hilsa population from becoming extinct, we equip our model with age-structured growth and harvest in hilsa and study the effect of environmental noise on both prey and predator populations. We formulate predator’s consumption using Beddington–DeAngelis functional response which also provides access to some alternative food source in the absence of prey. We obtain sufficient conditions for persistence in the mean, extinction, and global attractivity and prove stochastic ultimate boundedness and the existence of a unique positive periodic solution. We have included many numerical examples to validate the analytical results. Through global sensitivity analysis using the partial rank correlation coefficient technique, we observed the environmental noise to be of great influence on the prey–predator population.

In this research, we study and construct a dynamic prey-predator model. We include an element of intraspecific competition in both predators. We formulated the Holling type I response function for each predator. We consider all populations to be of economic value so that they can be harvested. We analyze the positive solution, the existence of the equilibrium points, and the stability of the balance points. We obtained the local stability condition by using the Routh-Hurwitz criterion approach. We also simulate the model. This research can be developed with different response function formulations and harvest optimization.

In this investigation, a predator-prey interaction model among Phytoplankton, Zooplankton and Fish has been developed. In the absence of Zooplankton and Fish, it is assumed that Phytoplankton grows logistically. It is assumed that Zooplankton consumes Phytoplankton and Fish consumes Phytoplankton as well as Zooplankton. Holling type I & II functional responses have been considered to formulate the our proposed model. It is considered that Phytoplankton releases some toxin in the aquatic environment which makes some death in Zooplankton population. Quadratic harvesting is considered on Fish species. Boundedness of the solution of our proposed model has also been studied. Local stability of the system around each equilibrium point has been investigated. Also, the global stability of the interior equilibrium point has been studied. Existence condition of Hopf bifurcation of our proposed system has been studied. It is found that half saturation constant (α) can change the system dynamics. It is also found that the harvesting rate of Fish (E) and consumption rate of Zooplankton (γ1) has a significant role in the stability of the system. Again, it is found that the harvesting of Fish species will be increased if the selling price of Fish (p) and the annual discount (δ1) of Fish production cost increases. It is also found that the optimal harvesting rate of Fish decreases due to the increase of cost (c) of harvesting of Fish. Finally, some numerical simulation results have been presented to verify our analytical findings.

A two-species predator–prey model where the predators are separated into juvenile and mature predators has been considered in this article. Here, functional response in linear form is considered to know the impact of adult predator-induced fear in the stage-structured prey–predator model. Positivity and boundedness of the model system have been checked. Persistence conditions and existence condition(s) of each equilibria have been derived. The local and global stability analysis have been implemented both analytically and numerically along with Hopf bifurcation analysis with direction. From the analysis of the model system, it is observed that the mature predator-induced fear and rate of transition from juvenile (minor) predator to mature predator plays a crucial role in controlling the system dynamics.

In this article, we investigate a fractional-order singular Leslie-Gower prey-predator bioeconomic model, which describes the interaction between populations of prey and predator, and takes into account the economic interest. We firstly obtain the solvability condition and the stability of the model system, and discuss the singularity induced bifurcation phenomenon. Next, we introduce a state feedback controller to eliminate the singularity induced bifurcation phenomenon, and discuss the optimal control problems. Finally, numerical solutions and their simulations are considered in order to illustrate the theoretical results and reveal the more complex dynamical behavior.

The objective of this paper is to study the dynamical properties of a discrete-time prey–predator model under imprecise biological parameters. We consider refuge for prey species as a constant number. The equilibria of the model are obtained, and the dynamic behaviors of the proposed system are examined for the interval biological parameters. Simulations of the model are performed for different parameters of the model. Numerical simulations demonstrate that the proposed discrete model exhibits rich dynamics of a chaotic and complex nature.

In this paper, we propose a fractional-order prey-predator model with infection on both populations. First, we prove some important results such as existence, uniqueness, non-negativity and bounded...

In this paper, a diffusive predator–prey model with Holling type II (Michaelis–Menten) functional response and a protection zone for prey is investigated. Dynamics and steady state solutions of the...

A prey-predator model with Michael Mentence type of predator harvesting and infectious disease in prey is studied. The existence, uniqueness and boundedness of the solution of the model are investigated. The dynamical behavior of the system is studied locally as well as globally. The persistence conditions of the system are established. Local bifurcation near each of the equilibrium points is investigated. Finally, numerical simulations are given to show our obtained analytical results.

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.