Narrow Results

In this paper, we consider the dynamics of a prey–predator model with logistic growth incorporating refuge in the prey and cooperation among predators population. Furthermore, multiplicative Allee effect in the prey growth is added to account from biological and mathematical perspectives. First, the existence and stability of equilibria of the model are discussed. Next, the existence of several kinds of bifurcation is provided and also studied the direction and stability of Hopf bifurcation. In addition, we study the influence of hunting cooperation on the model analytically and numerically, and find that the hunting cooperation cannot only reduce the density of prey population, but also destabilize the system dynamics irrespective of Allee effect. We choose also the impact of refuge on the model numerically, and explore that refuge stabilize the system dynamics. Moreover, the comparison between the dynamics of strong and weak Allee effect is taken into consideration.

In this paper, we present and assess a predator–prey Leslie–Gower model including disease, refuge and treatment in prey population. There are two groups of prey: those who are susceptible and infected. It is hypothesized that prey population is affected by diseases and refuge, and grows logistically in the absence of predators. Infected prey population receives treatment. The predators’ growth rate is governed by the modified Leslie–Gower dynamics. The dynamical attributes of the resulting system are boundedness, positivity of solutions, extinction criteria, existence and (local and global) stability. Biology uses mathematical analysis to identify the possible attributes of equilibrium points. The focus of this study is to assess how treatment and refuge affect the populations of ill prey, susceptible prey, predators and treated prey. The numerical simulation indicates that the influence of treatment, and refuge change the dynamics of the system (2.1). Extensive numerical simulations were performed to validate our analytical findings by using the Mathematica and MATLAB software.

This paper analyzes the generalization of a model presented in J. Bektešević, V. Hadžiabdić, S. Kalabušić, M. Mehuljić and E. Pilav [Dynamics of a class of host–parasitoid models with external stocking upon parasitoids, Adv. Differ. Equ.2021(31) (2021)]. The study explores the behavior of the solution near equilibrium points when the system has different outcomes, such as extinction, infinitely many exclusion points or unique exclusion and coexistence. We prove global stability for the extinction and host-exclusion equilibrium. We also investigate the non-hyperbolic case of parasitoid-exclusion equilibrium and delve deeper into the 1:1 resonance. The transcritical bifurcation occurs at the host-exclusion equilibrium, indicating a threshold for host population invasion through transcritical bifurcation. Moreover, the local dynamics around the coexisting equilibrium can be highly complex due to the appearance of the Neimark–Sacker and period-doubling bifurcations. We provide the explicit form of the first Lyapunov exponent for the Neimark–Sacker bifurcation. Through numerical examples, we illustrate the theoretical findings.

The present paper deals with a prey–predator model with prey refuge in proportion to both species, and the independent harvesting of each species. Our study shows that using refuge as control, it can break the limit cycle of the system and reach the required state of equilibrium level. We have established the optimal harvesting policy. The boundedness, feasibility of interior equilibria and bionomic equilibrium have been determined. The main observation is that the coefficient of refuge plays an important role in regulating the dynamics of the present system. Moreover, the variation of the coefficient of refuge changes the system from stable to unstable and vice-versa. Some numerical illustrations are given in order to support our analytical and theoretical findings.

Constant prey refuge with immigration and harvesting in two species would result in significant diversity in the dynamics of a prey–predator population. The phrase “refuge” increases the likelihood of prey population survival in the face of a predator population. Based on these findings, we created and examined a two-species prey–predator system with immigration and harvesting factors, including refuge to only prey population. All ecologically possible equilibrium points are studied for the proposed system. Routh–Hurwitz stability criterion is used for local stability analysis. Global stability of the interior equilibrium point is examined with a suitable Lyapunov function. Local bifurcation of the proposed system, such as saddle-node bifurcation, is analyzed. The conditions for the emergence of this bifurcation at the critical threshold near the nonhyperbolic equilibrium point are established by utilizing Sotomayor’s theorem. The transversality condition is validated for the occurrence of Hopf-bifurcation. The first Lyapunov number is exploited for determining the nature of Hopf bifurcating periodic solution. Finally, numerical simulations are illustrated to validate our theoretical predictions.

In this paper, a diffusive zooplankton–phytoplankton model with time delay and hunting cooperation is established. First, the existence of all positive equilibria and their local stability are proved when the system does not include time delay and diffusion. Then, the existence of Hopf bifurcation at the positive equilibrium is proved by taking time delay as the bifurcation parameter, and the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by using the center manifold theorem and the normal form theory in partial differential equations. Next, according to the theory of Turing bifurcation, the conditions for the occurrence of Turing bifurcation are obtained by taking the intraspecific competition rate of the prey as the bifurcation parameter. Furthermore, the corresponding amplitude equations are discussed by using the standard multi-scale analysis method. Finally, some numerical simulations are given to verify the theoretical results.

The objective of this work is to investigate the complex dynamics of a discrete predator–prey system using the method of piecewise constant argument for discretization. An analysis is conducted to examine the presence and stability of fixed points. Furthermore, the system is shown to undergo period-doubling (PD) and Neimark–Sacker (NS) bifurcations by the use of center manifold and bifurcation theories. The feedback and hybrid control strategies are used to regulate the system’s bifurcating and chaotic behaviors. Both strategies seem to be effective in managing bifurcation and chaos inside the system. Finally, the main results are validated by numerical evidence. Parameters of the system are varied to produce time graphs, phase portraits, bifurcation diagrams, and maximum Lyapunov exponent (MLE) graphs. The discrete model displays rich dynamics, as seen in the numerical simulations and graphs, indicating a complex and chaotic system.

In population biology, prey species try to maintain a delicate balance between reproduction and safety. The prey species subject to the mate-finding Allee effect may face a tough choice: risking life for more certain mating opportunities or decreasing mating activity to evade predators in the hope that a longer lifespan will compensate the reduced mating opportunities. In this paper, we consider a discrete-time predator–prey model where the logistically growing prey species experiences the mate-finding Allee effect. We assume that as predator density increases, prey individuals become more vigilant, leading to a decrease in their mate searching activity. We also consider that a constant proportion of prey individuals seek refuge from their predators due to the fear of predation risk. Our numerical simulations reveal that refuge has a stabilizing effect on the population dynamics, while for a certain amount of prey refuge, a decrease in the searching efficiency of mating partners due to the fear of predation risk has a destabilizing effect. We vary the parameters related to the predation-driven Allee effect and refuge simultaneously to explore system dynamics in the parameter plane and observe various intricate and striking properties of the system dynamics. The most significant discovery of our study is the novel organization of Arnold tongues in quasiperiodic and chaotic regimes. The organization of Arnold tongues looks like a “wing of a flapping bird”. Furthermore, we observe bistability between coexisting periodic attractors and notice intriguing patterns in the basins of attraction of these attractors. Lastly, we analyze the effect of the parameters related to the predation-driven Allee effect and refuge on the equilibrium densities of both species and also explore the dynamics of the analogous continuous-time system in the parameter plane.

Inverted biomass pyramids (IBPs) seem to be extremely rare in natural communities. Until recently, the only examples have been in freshwater and marine planktonic communities. In 2002 and 2008, investigators documented inverted biomass pyramids for nearly pristine coral reef ecosystems within the NW Hawaiian islands and the Line Islands, where apex predator abundance comprises up to 85% of the fish biomass. Large predator:prey biomass ratio seems to be a signature of nearly pristine coral reefs. While the mechanism responsible for the IBP for homogeneously mixed planktonic communities seems to be well understood, this mechanism is not strictly applicable to nearly pristine coral reefs where much of the prey use coral as refuge and are inaccessible to the predators. We construct a mathematical model with an explicit refuge to illustrate a new biologically plausible mechanism that can explain stable IBPs in nearly pristine coral reefs. New modeling components include a refuge of explicit size, a refuge size dependent functional response, and refuge size dependent prey growth rate. Utilizing realistic life history parameters of coral reef fishes, our model exhibits a stable inverted biomass pyramid. We prove that all fishing decreases the biomass ratio and sufficiently strong fishing transforms the inverted biomass pyramid to be bottom heavy. Finally we use our model to test the conjecture that pristine coral reefs will rebound faster from environmental shocks and find that it is not always true.

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results.

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.

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.

Fear of predation may assert privilege to prey species by restricting their exposure to potential predators, meanwhile it can also impose costs by constraining the exploration of optimal resources. A predator–prey model with the effect of fear, refuge, and hunting cooperation has been investigated in this paper. The system’s equilibria are obtained and their local stability behavior is discussed. The existence of Hopf-bifurcation is analytically shown by taking refuge as a bifurcation parameter. There are many ecological factors which are not instantaneous processes, and so, to make the system more realistic, we incorporate three discrete time delays: in the effect of fear, refuge and hunting cooperation, and analyze the delayed system for stability and bifurcation. Moreover, for environmental fluctuations, we further modify the delayed system by incorporating seasonality in the fear, refuge and cooperation. We have analyzed the seasonally forced delayed system for the existence of a positive periodic solution. In the support of analytical results, some numerical simulations are carried out. Sensitivity analysis is used to identify parameters having crucial impacts on the ecological balance of predator–prey interactions. We find that the rate of predation, fear, and hunting cooperation destabilizes the system, whereas prey refuge stabilizes the system. Time delay in the cooperation behavior generates irregular oscillations whereas delay in refuge stabilizes an otherwise unstable system. Seasonal variations in the level of fear and refuge generate higher periodic solutions and bursting patterns, respectively, which can be replaced by simple 1-periodic solution if the cooperation and fear are also allowed to vary with time in the former and latter situations. Higher periodicity and bursting patterns are also observed due to synergistic effects of delay and seasonality. Our results indicate that the combined effects of fear, refuge and hunting cooperation play a major role in maintaining a healthy ecological environment.

The diffusive Leslie–Gower predator–prey model incorporating a prey refuge is reconsidered here. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained, and our results supplement earlier ones.

In this paper, we propose a predator–prey system with a competitor for the prey. The model incorporates a constant prey refuge and predation process follows Holling type II response function. Using the Routh–Hurwitz criterion, the sufficient conditions of locally asymptotically stable of all the equilibria are obtained. Furthermore, global stability of the positive equilibrium is investigated by constructing a suitable Lyapunov function. The occurrence of Hopf-bifurcation of the system is shown at a critical value “$m$” and the system can be stabilized by increasing amount of prey refuge. The result includes the sufficient conditions for uniform persistence. Numerical simulations are carried out to illustrate the obtained results and the dependence of the dynamic behavior on the prey refuge.

In this paper, we analyze a system of delay differential equations incorporating prey’s refuge, fear, fear-response delay, extra food for predators and their gestation lag. First, we examined the system without delay. The persistence, stability (local and global) and various bifurcations are discussed. We provide detailed analysis for transcritical and Hopf-bifurcation. The existence of positive equilibria and the stability of prey-free equilibrium are interrelated. It is shown that (i) fear can stabilize or destabilize the system, (ii) prey refuge in a specific limit can be advantageous for both species, (iii) at a lower energy level (gained from extra food), the system undergoes a supercritical Hopf-bifurcation and (iv) when the predator gains high energy from extra food, it can survive through a homoclinic bifurcation, and prey may become extinct. The possible occurrence of bi-stability with or without delay is discussed. We observed switching of stability thrice via subcritical Hopf-bifurcation for fear-response delay. On changing some parametric values, the system undergoes a supercritical Hopf-bifurcation for both delay parameters. The delayed system undergoes the Hopf-bifurcation, so we can say that both delay parameters play a vital role in regulating the system’s dynamics. The analytical results obtained are verified with the numerical simulation.

The interaction of prey (small fish) and predator (large fish) in lakes/ ponds at temperate and tropical regions varies when water level fluctuates naturally during seasonal time. We relate the perceptible effect of fear and anti-predator behavior of prey with the water-level fluctuations and describe how these are influenced by the seasonal changing of water level. So, we consider these as time-dependent functions to make the system more realistic. Also, we incorporate the time-dependent delay in the negative growth rate of prey in predator–prey model with Crowley–Martin-type functional response. We clearly provide the basic dynamics of the system such as positiveness, permanence and nonpersistence. The existence of positive periodic solution is studied using Continuation theorem, and suffiecient conditions for globally attractivity of positive periodic solution are also derived. To make the system more comprehensive, we establish numerical simulations, and compare the dynamics of autonomous and nonautonomous systems in the absence as well as the presence of time delay. Our results show that seasonality and time delay create the occurrence of complex behavior such as prevalence of chaotic disorder which can be potentially suppressed by the cost of fear and prey refuge. Also, if time delay increases, then system leads a boundary periodic solution. Our findings assert that the predation, fear of predator and prey refuge are correlated with water-level variations, and give some reasonable biological interpretations for persistence as well as extinction of species due to water-level variations.

In this paper, we aim to study the effects of fear of predation in the presence of a competitor species and availability of refuge for the prey population. A discrete-time model of prey–predator–competitor type is formulated to study the dynamical relationship between the interacting species by deriving the conditions for local stability, permanence and existence of Neimark–Sacker bifurcation. A hybrid control technique based on state feedback strategy is implemented to control the chaos in the system. Finally, some numerical simulations are carried out to illustrate the theoretical discussions.