Search

This work investigates a prey–predator model featuring a Holling-type II functional response, in which the fear effect of predation on the prey species, as well as prey refuge, are considered. Specifically, the model assumes that the growth rate of the prey population decreases as a result of the fear of predators. Moreover, the detection of the predator by the prey species is subject to a delay known as the fear response delay, which is incorporated into the model. The paper establishes the preliminary conditions for the solution of the delayed model, including positivity, boundedness and permanence. The paper discusses the existence and stability of equilibrium points in the model. In particular, the paper considers the discrete delay as a bifurcation parameter, demonstrating that the system undergoes Hopf bifurcation at a critical value of the delay parameter. The direction and stability of periodic solutions are determined using central manifold and normal form theory. Additionally, the global stability of the model is established at axial and positive equilibrium points. An extensive numerical simulation is presented to validate the analytical findings, including the continuation of the equilibrium branch for positive equilibrium points. [ABSTRACT FROM AUTHOR]

This paper proposes a two-dimensional modified Leslie–Gower predator–prey model with carry-over effects, fear, prey protection and additional food, including prey harvesting. For this, it assumes the carry-over effect with delay incorporated in the birth rate of the prey population and gestation delay in Leslie–Gower term of a predator, and supposes that the predator consumes the prey according to Holling type II functional response. This dynamical system is split into two groups, namely non-delayed and delayed systems, respectively. In the non-delayed system, first, we discuss positivity, boundedness solutions, and the existence of equilibria, especially finding the coexistence of interior equilibrium points under certain conditions. After that, local and global stabilities are systematically analyzed. In addition, we obtain the conditions of Hopf-bifurcation in terms of the growth rate of prey species, prey protection and preference rate of predator for the non-delayed system. Whereas in the delayed system, both the delays have an essential role in governing the dynamical system discussed in three cases, and each case found the Hopf-bifurcation parameters for which the system changes stability behavior into instability. At last, the numerical simulations have been carried out to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator-prey diffusion model incorporating hunting cooperation and predator-taxis. Firstly, we establish the global existence of a classical solution for the model in any spatial dimension. Secondly, we analyze the stability/instability caused by predator-taxis, and we observe that predator-taxis play a key role in inducing stability changes. Specifically, if the positive equilibrium is stable for the corresponding reaction-diffusion model, the attractive predator-taxis can further stabilize the system, while the repulsive predator-taxis may lead to a change in spatial stability, if the positive equilibrium is unstable for the corresponding reaction-diffusion model, the attractive predator-taxis makes the model remain unstable, while the repulsive predator-taxis has a stabilizing effect. Finally, numerical simulations are employed to validate the obtained results. [ABSTRACT FROM AUTHOR]

This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model. [ABSTRACT FROM AUTHOR]

In this paper, we carry out some research on a predator–prey system with maturation delay, a stage structure for generalist predators and a Holling type-II functional response, which has already been proposed. First, for the delayed model, we obtain the conditions for the occurrence of stability switches of the positive equilibrium and possible Hopf bifurcation values owing to the growth of the value of the delay by applying the geometric criterion. It should be pointed out that when we suppose that the characteristic equation has a pair of imaginary roots λ = ± i ω (ω > 0) , we just need to consider i ω (ω > 0) due to the symmetry, which alleviates the computation requirements. Next, we investigate the nature of Hopf bifurcation. Finally, we conduct numerical simulations to verify the correctness of our findings. [ABSTRACT FROM AUTHOR]

The recent focus in the fields of biology and ecology has centered on the significant attention given to the mathematical modeling and analyzing the spatiotemporal population distribution among species engaged in interactions. This paper explores the dynamics of the temporal and spatiotemporal delayed Bazykin-type prey–predator model, incorporating fear and its carryover effect. In our model, we incorporated a functional response of the Holling-type II. In the temporal model, a detailed dynamic analysis was carried out, investigating the positivity and boundedness of solutions, establishing the uniqueness and existence of positive interior equilibria, and examining both local and global stability. Additionally, we explored the presence of saddle-node, transcritical, and Hopf bifurcations varying attack rate parameter. The delayed system shows highly periodic behavior. Additionally, for the spatiotemporal model, we provide a complete analysis of local and global stability, and we derive the conditions for the existence of Turing instability for both self-diffusion and cross-diffusion, respectively. The two-dimensional diffusive model is further discussed, highlighting various Turing patterns, including holes, stripes, and hot and cold spots, along with their biological significance. Numerical simulations are executed to validate the analytical findings in both temporal and spatiotemporal models. [ABSTRACT FROM AUTHOR]

This paper considers predator–prey systems in which the predator adapts the search speed to prey's density. The prey admits the cost of fear because of the predator and moves between source–sink patches to escape from predation. Using dynamical systems theory, we demonstrate equilibrium stability, uniform persistence and bifurcations in the system. It is shown that enhancing the search speed of predator and/or decreasing prey's dispersal from the source to sink could promote persistence of the system, while increasing the fear cost would stabilize the system. Moreover, dispersal could make the prey persist in both source–sink patches, even drive the predator into extinction. The dispersal could also make the prey reach a total population abundance larger than that without dispersal, even larger than the carrying capacity. Asymmetry in dispersal plays a role in the persistence and abundance. A novel finding of this work is that dispersal may lead to results reversing those without dispersal. Numerical simulations illustrate and extend our results. This work is important in understanding complexity in predation systems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a precise illustration for the structure of the nonconstant steady states for a Beddington–DeAngelis and Tanner predator–prey reaction–diffusion system with prey‐taxis. We treat the nonlinear prey‐taxis as a bifurcation parameter to analyze the bifurcation structure of the system. Furthermore, the exported global bifurcation theorem, under a rather natural condition, offers the existence of nonconstant steady states. In the proof, a priori estimates of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given. Finally, some conclusions including biological meanings are performed to summarize our main analytic results and future investigations. [ABSTRACT FROM AUTHOR]

This research paper investigates the numerical solutions of the predator–prey model through five recent numerical schemes (Adomian decomposition, El Kalla, cubic B‐spline, extended cubic B‐spline, exponential cubic B‐spline). We investigate the obtained computational solutions via the modified Khater methods. This model is considered as a well‐known bimathematical model to describe the prototype of an excitable system. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. The accuracy of the obtained numerical solutions is investigated by calculating the absolute error between the exact and numerical solutions. Many sketches are given to illustrate the matching between the exact and numerical solutions. [ABSTRACT FROM AUTHOR]

This paper aims to present a novel study on the dynamics of a fractional predator–prey–pathogen model to investigate the existence of the chaos in the model. We applied the Atangana–Baleanu fractional operator to the predator–prey–pathogen model, and new results are presented. Furthermore, the stability of the equilibrium points of the proposed model is investigated. The convergence and uniqueness of the solution for the model are also studied. Few numerical simulations have been performed for both predator, and prey populations. Some interesting chaotic behaviors of predator and prey populations of the model are also obtained by using an effective numerical scheme. Furthermore, corresponding numerical simulations were achieved for the various values of the fractional derivative. [ABSTRACT FROM AUTHOR]

This paper deals with a discrete diffusive predator–prey system involving two competing preys and one predator in a shifting habitat induced by the climate change. By applying Schauder's fixed‐point theorem on various invariant cones via constructing several pairs of generalized super‐ and subsolutions, we establish four different types of supercritical and critical forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two competing aboriginal coexistent preys with an invading alien predator, a pair of aboriginal coexistent predator–prey and an invading alien prey, and the coexistence of three species to the extinction state, respectively. Meanwhile, the nonexistence of some subcritical forced waves is showed by contradiction. Furthermore, some numerical simulations are given to present and promote the theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of kidnapping incidents through the application of the Lotka-Volterra model with fractional order modeling. Drawing parallels between predator–prey interactions in ecological systems and the relationship between Kidnappers and Victims, this study aims to provide insights into the complex phenomenon of kidnapping. Theoretical foundations are established through key definitions and theorems, paving the way for the construction of the model. The existence and uniqueness of solutions to the model equations are examined, followed by a stability analysis to assess the long-term dynamics of kidnapping incidents. Numerical simulations are conducted to validate the theoretical framework and explore the sensitivity of the model to different parameters and initial conditions. By employing the Lotka-Volterra model with fractional order modeling, the study gives a deeper understanding of kidnapping incidents by drawing parallels with ecological systems, allowing for a more comprehensive analysis of the complex interactions between perpetrators and victims. [ABSTRACT FROM AUTHOR]