Narrow Results

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.

In this article, a prey-predator system with Holling type II functional response for the predator population including prey refuge region has been analyzed. Also a harvesting effort has been considered for the predator population. The density-dependent mortality rate for the prey, predator and super predator has been considered. The equilibria of the proposed system have been determined. Local and global stabilities for the system have been discussed. We have used the analytic approach to derive the global asymptotic stabilities of the system. The maximal predator per capita consumption rate has been considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighborhood of interior equilibrium point. Also, we have used fishing effort to harvest predator population of the system as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent is earned from the resource. Finally, we have presented some numerical simulations to verify the analytic results and the system has been analyzed through graphical illustrations.

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.

The predator–prey model is fundamentally important to study the growth law of the population in nature. In this paper, we propose a diffusive predator–prey model, in which we also consider time delay in the gestation time of predator and Michaelis–Menten type predator harvesting. By analyzing the distribution of eigenvalues, we investigate the stability of the coexisting equilibrium and the existence of Hopf bifurcation using time delay as bifurcation parameter. We analyze the property of Hopf bifurcation, and give an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, some numerical simulations are given to support our results.

In this paper, a phytoplankton–zooplankton model incorporating toxic substances and nonlinear phytoplankton harvesting is established. The existence and stability of the equilibrium of this model are first investigated. The occurrence of transcritical, saddle-node, Hopf and Bautin bifurcations at different equilibria is then verified. In addition, the properties of Hopf bifurcation and Bautin bifurcation are discussed by using normal form method. These results demonstrate that phytoplankton and zooplankton populations will oscillate periodically when the harvesting level is high. More interestingly, it is found that the oscillations are always unstable for small phytoplankton carrying capacity, while the dynamics have close relations with the initial population densities for a large environmental capacity. The existence of Bautin bifurcation theoretically indicates that toxic phytoplankton can cause extinction once there exist harmful algal blooms for some time. These results are numerically illustrated for the model with spatial diffusion, which shows that local phytoplankton blooms will lead to global populations extinction.

In this manuscript, we consider an extended version of the prey–predator system with nonlinear harvesting [Gupta et al., 2015] by introducing a top predator (omnivore) which feeds on more than one trophic levels. Consideration of third species as omnivore makes the system a food web of three populations. We have guaranteed positivity as well as the boundedness of solutions of the proposed system. We observed that the presence of third species complicates the dynamical behavior of the system. It is also observed that multiple positive steady states exist for the proposed system which makes the problem more interesting compared to the similar models studied previously. Sotomayor’s theorem is being utilized to study the saddle-node bifurcation. The persistence conditions are discussed for the proposed model. The local existence of periodic solution through Hopf bifurcations is also guaranteed numerically. It is observed that the proposed model is capable to exhibit more complicated dynamics in the form of chaos in both the cases when there are unique and multiple coexisting steady states. Bifurcation diagrams and Lyapunov exponents have been drawn to ensure the existence of chaotic dynamics of the system.

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.

This study discusses the dynamic behaviors of the prey–predator model subject to the Allee effect and the harvesting of prey species. The existence of fixed points and the topological categorization of the co-existing fixed point of the model are determined. It is shown that the discrete-time prey–predator model can undergo Flip and Neimark–Sacker bifurcations under some parametric assumptions using bifurcation theory and the center manifold theorem. A chaos control technique called the feedback-control method is utilized to eliminate chaos. Numerical examples are given to support the theoretical findings and investigate chaos strategies’ effectiveness and feasibility. Additionally, bifurcation diagrams, phase portraits, maximum Lyapunov exponents, and a graph showing chaos control are demonstrated.

In this paper, we investigate a predator–prey model with Holling-II functional response, Allee effect and constant-yield predator harvesting, by comparing the differences between $h_1=0$ and $h_1>0$, where $h_1$ denotes the harvesting rate of predators. For system without harvesting, the Allee effect leads to population extinction. The system has at most one positive equilibrium and has a supercritical Hopf bifurcation which depends on the natural mortality rate of predators. Besides, by using normal form theory, we show that the system with $h_1>0$ reveals rich dynamic properties, including saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, where numerical simulations are presented to demonstrate the Bogdanov–Takens bifurcation of codimension 2 with a limit cycle and a homoclinic cycle. The system can generate up to two positive equilibria with the changes of $h_1$, which indicates that appropriate predator harvesting can assist in regulating the ecosystem. We then give the optimal harvesting strategy by using Pontryagin’s maximum principle. Finally, numerical simulations are performed to validate the functions of Allee effect and harvesting. Theoretical studies and numerical simulations demonstrate that the Allee effect can lead to species extinction and highlight the role of appropriate harvesting in controlling the stability of the system.

This paper deals with the dynamic behaviors of a discrete-time fractional-order predator–prey model in the presence of both the Allee effect and immigration on the prey population and in the presence of harvesting on the predator population. The existence and uniqueness and parametric conditions for local asymptotic stability of fixed points of the discrete-time fractional-order model are studied. Moreover, using the center manifold theorem and bifurcation theory, it is shown that the considered model undergoes flip and Neimark–Sacker bifurcations in a small neighborhood of the interior fixed point. Then, the direction of bifurcation is calculated. A feedback controller is implemented in the proposed model to control chaos thanks to the emergence of the Neimark–Sacker bifurcation. Furthermore, numerical analysis confirms the theoretical analysis with the help of Matlab software.

In this paper we have considered a prey-predator model with Holling type of predation and independent harvesting in either species. The purpose of the work is to offer mathematical analysis of the model and to discuss some significant qualitative results that are expected to arise from the interplay of biological forces. Our study shows that, using the harvesting efforts as controls, it is possible to break the cyclic behaviour of the system and drive it to a required state. Also it is possible to introduce globally stable limit cycle in the system using the above controls.

In a fully dynamic model of an open-access fishery, the level of fishery effort expands or contracts according as the net economic revenue (i.e., perceived rent) to the fisherman is positive or negative. A model reflecting this dynamic interaction between the perceived rent and the effort in a fishery, is called a dynamic reaction model. In this paper, we study a dynamic reaction model, in which the prey species is subjected to harvesting in the presence of a predator and a tax. It is also assumed that the gross rate of investment of capital in the fishery is proportional to the perceived rent. With this capital theoretic approach, the dynamical system consisting of the growth equations of the two species and also of the fishing effort is formulated. The steady state is determined and its stability is discussed. The object is to maximize the monetary social benefit as well as prevent the predator from extinction, keeping the ecological balance.

An analysis is presented for a model of a two species prey-predator system subject to the combined effects of delay and harvesting. Our study shows that, both the delay and harvesting effort may play a significant role on the stability of the system. Computer simulations are carried out to explain some of the mathematical conclusions.

This article concentrates on the study of delay effect on a model of schistosomiasis transmission with control measures such as predation or harvesting and chemotherapy. In the presence of predation or harvesting and chemotherapy, system admits multiple endemic equilibria. Mathematical analysis shows that they are opposite in nature regarding stability. One may observe switching phenomena for the unstable equilibrium by incorporating delay. The disease may be highly endemic if there is no control measure, which is obvious from the model analysis. Results obtained in this paper are also verified through numerical simulations.

A prey-predator model with stage structure for prey and selective harvest effort on predator is proposed, in which gestation delay is considered and taxation is used as a control instrument to protect the population from overexploitation. It is established that when the discrete time delay is zero, the model system is stable around the interior equilibrium and an optimal harvesting policy is discussed with the help of Pontryagin's maximum principle; On the other hand, stability switch of the model system due to the variation of discrete time delay is also studied, which reveals that the discrete time delay has a destabilizing effect. As the discrete time delay increases through a certain threshold, a phenomenon of Hopf bifurcation occurs and a limit cycle corresponding to the periodic solution of model system is also observed. Numerical simulations are carried out to show the consistency with theoretical analysis.

In this paper a predator-prey model with discrete delay and harvesting of predator is proposed and analyzed by considering ratio-dependent functional response. Conditions of existence of various equilibria and their stability have been discussed. By taking delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. Numerical simulations are also performed to illustrate the results.

This paper studies a differential-algebraic predator–prey system with prey harvesting, which consists of two differential equations and an algebraic equation. By using the differential-algebraic system theory, bifurcation theory and formal series expansions, we investigate the Hopf bifurcation and center stability of the differential-algebraic predator–prey system. Some sufficient conditions on these issues are obtained. In addition, numerical simulations illustrate the effectiveness of our results and their biological implications are discussed.

The progressive and increasing invasion of an opportunistic predator, the lionfish (Pterois volitans) has become a major threat for the delicate coral-reef ecosystem. The herbivore fish populations, in particular of Parrotfish, are taking the consequences of the lionfish invasion and then their control function on macro-algae growth is threatened. In this paper, we developed and analyzed a stage-structured mathematical model including P. volitans (lionfish), a cannibalistic predator, and a Parrotfish, its potential prey. As control upon the over predation, a rational harvest term has been considered. Further, to make the system more realistic, a delay in the growth rate of juvenile P. volitans population has been incorporated. We performed a global sensitivity analysis to identify important parameters of the system having significant correlations with the fishes. We observed that the system generates transcritical bifurcation, which takes the P. volitans-free equilibrium to the coexistence equilibrium on increasing the values of predation rate of adult P. volitans on Parrotfish. Further increase in the values of the predation rate of adult P. volitans on Parrotfish drives the system into Hopf bifurcation, which induces oscillation around the coexistence equilibrium. Moreover, the conversion efficiency due to cannibalism also has the property to alter the stability behavior of the system through Hopf bifurcation. The effect of time delay on the dynamics of the system is extensively studied and it is observed that the system develops chaotic dynamics through period-doubling oscillations for large values of time delay. However, if the system is already oscillatory, then the large values of time delay causes extinction of P. volitans from the system. To illustrate the occurrence of chaotic dynamics in the system, we drew the Poincaré map and also computed the Lyapunov exponents.

In this paper, a three-species food chain model has been developed by considering the interaction between prey, predator and super-predator species. It is assumed that in the absence of predator and super-predator species, the prey species grow logistically. It is also assumed that predator and super-predator consume prey and predator, respectively. It is assumed that the predator shows refuge behavior to the super-predator. Again, harvesting of super-predator population has been considered. It is assumed that the consumption of prey and predator follows Crowley–Martin-type functional form. Boundedness of the solution of the system has been studied and different equilibrium points are determined and the stability of the system around these equilibrium points has been investigated. Existence conditions of Hopf bifurcation with respect to ${\gamma }_1$ of the system have been studied. It is found that the system shows some complex and critical dynamics due to increase of handling time of prey. It is also found that the system moves towards stable steady state due to increase of predator interference. It is observed that predator refuge may be responsible for the stability of the system. The chaotic dynamics of the system have been found due to the increase of the harvesting rate of super-predator.

In this paper, we investigate the dynamics of a system in which both prey and predator are harvested with constant rate. Our main objective is to find the effects of harvesting on equilibria, stability, and bifurcations in the system, which may be useful for biological management. The existence and stability of equilibrium points of the model are further investigated. A thorough qualitative analysis has been carried out based on bifurcation theory in dynamical systems and to validate our analytical findings, a large scale numerical simulation has been performed by using plausible values of parameters involved. It is shown that the model can exhibit Hopf bifurcation. The first Lyapunov coefficient is calculated to determine the direction of limit cycle of Hopf bifurcation. Also, it has been proven analytically that the system exhibits Bogdanov–Takens bifurcation of codimension 2. Moreover, discrete-time delay effect has been included due to gestation of the predator species on the same system and observed Hopf bifurcation with respect to the delay parameter. This study renders important tools for investigations of the dynamics of biotic organisms for the management and control of over harvesting. Some phase plane analysis has been carried out to support our analytical results.