Interior Equilibrium Research Articles

Local dynamics of a predator–prey community in a moderate period of time

In this work, we have introduced an ecological model of a prey–predator system. It is assumed that the prey species grows logistically, but the total number of predator is constant in the time interval. Positivity and boundedness of the solution ensure that the proposed model is well-posed. Local stability conditions of the equilibrium points have been analysed by the Routh–Hurwitz criterion. The persistence of the system has also been shown under a parametric restriction. Numerical analysis has indicated that both axial and interior steady states can exist only for moderate consumption rate (searching efficiency). But if this rate becomes high (or low), then only the prey-free equilibrium (or one of the interior equilibriums) exists as a steady state. Further, the equilibrium points can change their stability through transcritical and saddle-node bifurcations by varying the consumption rate of the predator. Analytical results provide an interesting phenomenon about this model: the system can never show any oscillating behaviour for any parametric values, i.e. no limit cycle can occur through Hopf bifurcation around an equilibrium point. The axial equilibrium becomes stable from an unstable situation when the consumption rate becomes high and the interior state which is stable remains stable as time goes by.

Preservation of the Forestry Biomass and Control of Increasing Atmospheric $$\hbox {CO}_2$$ using Concept of Reserved Forestry Biomass

This paper deals with a non-linear mathematical model which is formulated by introducing the concept of reserved forestry biomass. The aim of the proposed mathematical model is to study a dynamical system consisting of the human population with reserved and unreserved forestry biomasses, and the increasing concentration of carbon dioxide in the atmosphere. Conditions for existence, uniqueness, and local stabilities of all possible equilibria are derived and ecologically explained. The conditions for global stability of the interior equilibrium point and uniform persistence of the system, are derived. Numerical simulations are also performed for further studies of the mathematical model. The analysis clearly shows that too much exploitation of forestry biomass results in their extinction and a huge increase in the concentration of atmospheric carbon dioxide. From the analysis, we have observed that the reserved forestry biomass is more effective in controlling the level of $$\hbox {CO}_2$$ in the atmosphere than the unreserved forestry biomass. Hence, by this study, we convey the message that besides the preservation of forestry biomasses, reserved forestry biomasses may play a significant role to save our environment too.

Bimatrix Replicator Dynamics with Periodic Impulses

This paper investigates the bimatrix replicator dynamics with periodic impulses. In biological system, impulsive perturbations may due to the occurrence of an unfavorable physical environment, or due to the seasonal life history effects in the physiological and reproductive mechanisms of the population. We show that impulsive perturbations can lead to complicated dynamical behaviors. On the one hand, the system can have multiple $$\tau $$ -periodic solutions, where the lower bound of the number of solutions is increasing linearly in the impulsive period $$\tau $$ . On the other hand, for shorter impulsive period, we provide a differential approximation for the impulsive dynamical system. By analyzing the resulting differential dynamics, we show that the interior equilibrium (which corresponds to a $$\tau $$ -periodic solution) must be globally stable if it exists. Furthermore, when the impulsive effect is weak, all interior trajectories of the impulsive dynamics evolve to a small neighborhood of the interior equilibrium of the bimatrix replicator dynamics. In summary, longer impulsive period causes an increase in the complexity of the evolutionary process and shorter period promotes evolutionary stability of the interior equilibrium where multiple strategies can coexist.

Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow.

The mathematical analysis of a three-tiered food-web describing anaerobic chlorophenol mineralisation has suggested the emergence of interesting dynamical behaviour through its specific ecological interactions, which include competition, syntrophy, and product inhibition. Previous numerical analyses have revealed the possibility of a Hopf bifurcation at the interior equilibrium as well as the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired operating region where the positive steady-state is stable and full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and show the occurrence of a supercritical Hopf bifurcation resulting from variations in these operating parameters. Further, using numerical estimation, we also show that variations in the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations. Finally, we are able to apply persistence theory for a range of parameter sets to demonstrate persistence in cases where chlorophenol and hydrogen are extraneously added to the system, mirroring recent applied studies highlighting the role of hydrogen in maintaining stable anaerobic microbial communities.

Dynamics of a toxin-mediated aquatic population model with delayed toxic responses.

Traditional mathematical models for studying the effects of environmental toxins on population dynamics usually assume that the toxic effects are immediate. However, the effects of toxins on the reproduction and mortality of aquatic populations can be delayed in reality. In this paper, we propose a new mathematical model with delayed toxic responses for aquatic populations in polluted aquatic environments. The delayed model is analyzed in terms of steady states, stability, and bifurcation. The results show that the delayed effect on the reproduction of aquatic populations does not affect the stability of the interior equilibrium, but the delayed effect on the mortality of aquatic populations can destabilize the interior equilibrium. Numerical results corroborate the theoretical ones are presented.

Global dynamics of HIV infection with two disease transmission routes - a mathematical model

In this paper, we have studied the global dynamics of HIV model with two transmission paths: direct transmission through cells-to-cells contact and indirect transmission through virus. We have derived a four dimensional mathematical model including uninfected $CD_4^{+T}$ cells, infected $CD4^+{T}$ cells, virus and the CTL immune response cells. The nonnegativity and boundedness property of the solutions the proposed mathematical system have been analysed, and the basic reproduction ratio $R_0$ has been derived with the help of next generation matrix method. We also discussed the local and global stability with respect to the basic reproduction ratio of both disease-free and interior equilibrium points under certain conditions. Through numerical simulations, we have validated the all analytical findings. We have established that the disease-free equilibrium is globally stable for $R_0 1$ whenever exists. It is also observed that cells-to-cells transmission rate is more effective compare to virus-to-cells infection rate.

Global Asymptotic Stability and Hopf Bifurcation in a Homogeneous Diffusive Predator-Prey System with Holling Type II Functional Response

In this paper, we considered a homogeneous reaction-diffusion predator-prey system with Holling type II functional response subject to Neumann boundary conditions. Some new sufficient conditions were analytically established to ensure that this system has globally asymptotically stable equilibria and Hopf bifurcation surrounding interior equilibrium. In the analysis of Hopf bifurcation, based on the phenomenon of Turing instability and well-done conditions, the system undergoes a Hopf bifurcation and an example incorporating with numerical simulations to support the existence of Hopf bifurcation is presented. We also derived a useful algorithm for determining direction of Hopf bifurcation and stability of bifurcating periodic solutions correspond to j ≠ 0 and j = 0, respectively. Finally, all these theoretical results are expected to be useful in the future study of dynamical complexity of ecological environment.

Dynamics of a stage-structured population model with a state-dependent delay

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough.

Bifurcation analysis of a Singular Nutrient-plankton-fish model with taxation, protected zone and multiple delays

A differential algebraic nutrient-plankton-fish model with taxation, free fishing zone, protected zone and multiple delays is investigated in this paper. First, the conditions of existence and control of singularity induced bifurcation are given by regarding economic interest as bifurcation parameter. Meanwhile, the existence of Hopf bifurcations are investigated when migration rates, taxation and the cost per unit harvest are taken as bifurcation parameters respectively. Next, the local stability of the interior equilibrium, existence and properties of Hopf bifurcation are discussed in the different cases of five delays. Furthermore, the optimal tax policy is obtained by using Pontryagin's maximum principle. Finally, some numerical simulations are presented to demonstrate analytical results.

Harvesting in tri-trophic food chain stabilises the chaotic dynamics-conclusion drawn from Hastings and Powell model

The paper explores a tri-trophic food chain model with harvesting in the species. The curiosity of this paper is to observe chaotic dynamics and its control. We perform the local stability analysis of the equilibrium points. The Hopf bifurcation analysis and global stability around the interior equilibrium point are also performed. Our numerical simulations reveal that the three species food chain model induces chaos from period-doubling, limit cycle and stable focus for increasing values of half saturation constant. We conclude that chaotic dynamics can be controlled by the harvesting parameter. We apply basic tools of non-linear dynamics such as Poincare section and Lyapunov exponent to identify chaotic behaviour of the system.

Harvesting in tri-trophic food chain stabilises the chaotic dynamics-conclusion drawn from Hastings and Powell model

The paper explores a tri-trophic food chain model with harvesting in the species. The curiosity of this paper is to observe chaotic dynamics and its control. We perform the local stability analysis of the equilibrium points. The Hopf bifurcation analysis and global stability around the interior equilibrium point are also performed. Our numerical simulations reveal that the three species food chain model induces chaos from period-doubling, limit cycle and stable focus for increasing values of half saturation constant. We conclude that chaotic dynamics can be controlled by the harvesting parameter. We apply basic tools of non-linear dynamics such as Poincare section and Lyapunov exponent to identify chaotic behaviour of the system.

The Dynamical Behavior of a Predator-Prey System with Holling Type II Functional Response and Allee Effect

In this paper, we mainly considered the dynamical behavior of a predator-prey system with Holling type II functional response and Allee-like effect on predator, including stability analysis of equilibria and Hopf bifurcation. Firstly, we gave some sufficient conditions to guarantee the existence, the local and global stability of equilibria as well as non-existence of limit cycles. By using the cobweb model, some cases about the existence of interior equilibrium are also illustrated with numerical outcomes. These existence and stability conclusions of interior equilibrium are also suitable in corresponding homogeneous reaction-diffusion system subject to the Neumann boundary conditions. Secondly, we theoretically deduced that our system has saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation under certain conditions. Finally, for the Hopf bifurcation, we choose d as the bifurcation parameter and presented some numerical simulations to verify feasibility and effectiveness of the theoretical derivation corresponding to the existence of yk, respectively. The Hopf bifurcations are supercritical and limit cycles generated by the critical points are stable.

Delay induced multiple stability switch and chaos in a predator–prey model with fear effect

We propose a delayed predator–prey model with fear in the prey population. We consider that the growth rate of the prey population is suppressed due to the fear of predators. It is also considered that there is a time lag between the time of perceiving predator signals through chemical and/or vocal cues and the changes in life-history and behavioral responses in the prey population. We study boundedness, persistence, local and global behavior of the delayed system. Moreover, the Hopf-bifurcation analysis around the interior equilibrium with respect to the delay parameter is established. The stability and direction of Hopf-bifurcation are also studied. It is observed that fear induced delay has both stabilizing and destabilizing effects depending on the magnitude of the delay parameter. We observe that for the gradual increase of the magnitude of delay, the system dynamics switches multiple times between stable focus and limit cycle oscillations. However, for a higher value of the delay parameter, the system ultimately enters into the chaotic regime. The delay system also exhibits node-cycle bi-stability behavior between the interior equilibrium point and stable limit cycle. Numerical simulations are also performed to validate analytical findings.

Flip bifurcation and Neimark–Sacker bifurcation in a discrete predator–prey model with harvesting

In this paper, a difference-algebraic predator–prey model is proposed, and its complex dynamical behaviors are analyzed. The model is a discrete singular system, which is obtained by using Euler scheme to discretize a differential-algebraic predator–prey model with harvesting that we establish. Firstly, the local stability of the interior equilibrium point of proposed model is investigated on the basis of discrete dynamical system theory. Further, by applying the new normal form of difference-algebraic equations, center manifold theory and bifurcation theory, the Flip bifurcation and Neimark–Sacker bifurcation around the interior equilibrium point are studied, where the step size is treated as the variable bifurcation parameter. Lastly, with the help of Matlab software, some numerical simulations are performed not only to validate our theoretical results, but also to show the abundant dynamical behaviors, such as period-doubling bifurcations, period 2, 4, 8, and 16 orbits, invariant closed curve, and chaotic sets. In particular, the corresponding maximum Lyapunov exponents are numerically calculated to corroborate the bifurcation and chaotic behaviors.

Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems

Recent demographic experiments have demonstrated that both birth and survival in free-living animals are essentially affected due to having sufficient exposure to predators and further leaving physiological stress effects. In this paper, we have proposed and analyzed a predator–prey interaction model with Beddington–DeAngelis functional response (BDFR) and incorporating the cost of fear into prey reproduction. Stability analysis and the existence of transcritical bifurcation are studied. For the spatial system, the Hopf-bifurcation around the interior equilibrium, stability of homogeneous steady state, direction and stability of spatially homogeneous periodic orbits have been established. Using Normal form of the steady state bifurcation, the possibility of pitchfork bifurcation has been established. The impact of the level of fear and mutual interference on the stability and Turing patterns of the spatiotemporal system have been discussed in detail. Simulation results ensure that the fear of predator stabilizes the system dynamics and cost the overall population size of the species.

Bifurcation in Plant-Pest-Natural Enemy Interaction Dynamics with Gestation Delay for Both Pest and Natural Enemy

During this analysis, as per natural control approach in pest management, a plant-pest dynamics with biological control is proposed, here assuming that the pest and natural enemy are having different levels of gestation delay and harvesting rate of pests by natural enemy follows Holling type-III response function. Boundedness and positivity of the system are studied. Equilibria and stability analysis is carried out for possible equilibrium points. The existence of Hopf bifurcation at interior equilibrium is presented. The sensitivity analysis of the system at interior equilibrium point for model parameters has been explored. Numerical simulations are performed to support our analytic findings.

Stability and Numerical Simulation of Prey-predator System with Holling Type-II Functional Responses for Adult Prey

In this paper, we analyze the local stability of the prey-predator model. This model was constructed from two prey involving stage-structure and one predator. The population of prey is divided into two subpopulations, adult prey and immature. The ratio of intrinsic growth of predator population density divided by adult prey conversion factors into a predator. Reduction of predator populations has a reciprocal relationship with the availability of favorite foods determined by the environmental resources. We also consider that Holling type II the functional response for the predator. We have investigated the local stability and the solution of the system. We have studied the existence and feasibility of various equilibrium points. The system has three positive equilibria, namely the original, the extinction of the predator and the interior point. We explore the stabilities of all nonnegative equilibrium point by the real parts of the eigenvalues of the Jacobian matrix at each equilibrium point must be negative. We also use the Routh-Hurwitz criterion to investigate the stability at the interior equilibrium point. This study aims to detect the limit cycle with its phase portrait and its time-series graphs of the prey-predator system. Numerical simulated are represented as phase portraits to draw the stability of the equilibrium point. In our Numerical outcomes confirmed the analytical results and the local behavior prey-predator model which start from their positive initial condition. The parameter value is taken mainly from the literature and assumption. By a computational Python using the fourth-order Runge-Kutta method, we solved the prey-predator system and showed some new phase portraits such as the existence of the stable or unstable equilibrium point under a suitable value of the parameter. The next research, we will explore the global stability analysis behavior of the interior equilibrium point.

Spatiotemporal Dynamics of a Diffusive Predator–Prey System with Allee Effect and Threshold Hunting

In this paper, we study a diffusive predator–prey system with the Allee effect and threshold hunting. First, the number of interior equilibrium points is determined by discussing the relation of parameters. Then, preliminary analysis on the local asymptotic stability and bifurcations of non-spatial system based on ordinary differential equations is presented. It is noted that four stable equilibrium points coexist due to the Allee effect and threshold hunting. The stability of interior equilibrium points and the existence of Turing instability induced by the diffusion, spatially homogeneous and inhomogeneous Hopf bifurcation, Turing–Hopf bifurcation are studied by analyzing the corresponding characteristic equation for spatial system. By constructing generalized Jacobian matrix, we analyze the stability of interior equilibrium point where u-component is equal to the threshold of functional response. These results show that the Allee effect, threshold hunting and diffusion have significant impacts on the dynamics. Last, we present some numerical simulations that supplement the analytic results.

An Application of a Delay CSOH Model on the Coconut Farm

In this paper, we introduce the mathematical model that represents the quantity and population dynamics on the coconut farm. The model encompasses the number of coconuts and population of squirrels, barn owls, and squirrel hunters. We study the fundamental properties of the model that include positivity, boundedness, and equilibrium points. We also investigate the effect of the time delay on the stability of the equilibrium points. The results of the analysis show that when the time delay reaches its critical value, the interior equilibrium point lost its stability, and there occurs the Hopf bifurcation.

Credit Attribution and Collaborative Work

We examine a dynamic model of teamwork in which the public attributes credit for success based on its perception of individual efforts. The collaborative behavior varies starkly depending on the shape of marginal effort cost, or project’s “difficulty.” In the unique (interior) equilibrium, higher ability collaborators work less and thus receive lower credit and payoff for “easy” projects, while the reverse holds for “difficult” projects. Despite free-riding, the team equilibrium may involve over-investment. Social efficiency requires over-rewarding collaborative work and under-rewarding solo work. The incentives to team up and the impact of effort monitoring on credit attribution are also investigated.