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The anti‐predator decision making may lower down/change the food/energy intake of prey which naturally translate into a decrease in their reproductive output. More importantly, experimental studies also ensure that the predator‐induced fear (indirect effect) may alter much behavior of the prey even more strongly than the effect of direct killing via habitat changes, vigilance, foraging, and different physiological changes. Considering this fact, we analyze a delayed predator‐prey system incorporating anti‐predator defense. The interaction term has been modeled using Beddington‐DeAngelis functional response (BDFR). We discuss the local and global stability, permanence, and non‐permanence to comprehend the dynamics of the system. By constructing a suitable Lyapunov function, globally asymptotic stability has been established. We also demonstrate that a certain value of delay may cause Hopf‐bifurcation and period‐doubling route to chaos. It is shown that under certain parametric conditions, the gestation delay‐induced oscillations could be prevented. Using Nyquist criterion, the length of delay has been estimated for which the stability continues to hold. Moreover, we find that spatial diffusion affects the critical value of the bifurcation parameter and demonstrate the existence of spatially homogeneous periodic solutions with the change of time delay. In addition, the level of fear also affects the dynamical behaviors of the delayed system with/without diffusion. These results indicate that the gestation delay induces oscillatory coexistence mode of the interaction between predator and its prey, which reveals the driving factors for the co‐existence of predator and prey in the form of periodic oscillations in actual ecosystems. [ABSTRACT FROM AUTHOR]

Consider the heterogeneity (e.g., heterogeneous social behaviour, heterogeneity due to different geography, contrasting contact patterns and different numbers of sexual partners etc.) of host population, in this paper, the authors propose an infection age multi-group SEIR epidemic model. The model system also incorporates the feedback variables, where the infectivity of infected individuals may depend on the infection age. In the direction of mathematical analysis of model, the basic reproduction number R0 has been computed. The global stability of disease-free equilibrium and endemic equilibrium have been established in the term of R0. More precisely, for R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable and for R0 > 1, they establish global stability of endemic equilibrium using some graph theoretic techniques to Lyapunov function method. By considering a numerical example, they investigate the effects of infection age and feedback on the prevalence of the disease. Their result shows that feedback parameters have different and even opposite effects on different groups. However, by choosing an appropriate value of feedback parameters, the disease could be eradicated or maintained at endemic level. Besides, the infection age of infected individuals may also change the behaviour of the disease, global stable to damped oscillations or damped oscillations to global stable. [ABSTRACT FROM AUTHOR]

• We construct a Leslie-Gower type prey-predator system with feedback. • We systematically analyze the effects of feedback controls on the dynamics of ecosystems. • Pattern transition emerges as feedback intensity varies. In this study, we investigate the global dynamics of non-autonomous and autonomous systems based on the Leslie–Gower type model using the Beddington–DeAngelis functional response (BDFR) with time-independent and time-dependent model parameters. Unpredictable disturbances are introduced in the forms of feedback control variables. BDFR explains the feeding rate of the predator as functions of both the predator and prey densities. The global stability of the unique positive equilibrium solution of the autonomous model is determined by defining an appropriate Lyapunov function. The condition obtained for the global stability of the interior equilibrium ensures that the global stability is free from control variables, which is also a significant issue in the ecological balance control procedure. The autonomous system exhibits complex dynamics via bifurcation scenarios, such as period doubling bifurcation. We prove the existence of a globally stable almost periodic solution of the associated non-autonomous model. The different coefficients of the system are taken as almost periodic functions by generalizing periodic assumptions. The permanence of the non-autonomous system is established by defining upper and lower averages of a function. Our results also explain how the important hypothesis in ecology known as the "intermediate disturbance hypothesis" applies in predator–prey interactions. We show that moderate feedback intensity can make both the ordinary differential equation system and partial differential equation system more robust. The results obtained provide new insights into the protection of populations, where moderate feedback intensity can promote the coexistence of species and adjusting the intensity of the feedback in appropriate regions can control the population biomass while maintaining the stability of the system. Finally, the results obtained from extensive numerical simulations support the analytical results as well as the usefulness of the present study in terms of ecological balance and bio-control problems in agro-ecosystems. [ABSTRACT FROM AUTHOR]

In this paper, an attempt has been made to understand the role of predator's interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green's first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots → stripes → spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established. [ABSTRACT FROM AUTHOR]

This paper discusses a predator-prey model with prey refuge. We investigate the role of prey refuge on the existence and stability of the positive equilibrium. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional, which shows that the prey refuge has no influence on the permanence property of the system. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. To access the usability of proposed predator-prey model in practical scenarios, we also suggest, the use of Levenberg-Marquardt (LM) method for associated parameter estimation problem. Numerical results demonstrate faithful reconstruction of system dynamics by estimated parameter by LM method. The analytical results found in this paper are illustrated with the help of suitable numerical examples. [ABSTRACT FROM AUTHOR]