Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior

In this study, we consider a diffusive predator–prey model with multiple Allee effects induced by fear factors. We investigate the existence, boundedness and permanence of the solution of the system. We also discuss the existence and non-existence of non-constant solutions. We derive sufficient conditions for spatially homogeneous (non-homogenous) Hopf bifurcation and steady state bifurcation. Theoretical and numerical simulations show that strong Allee effect and fear effect have great effect on the dynamics of system.

In this paper, we investigate the effect of nonlocal delay on stability and spatiotemporal dynamics of a diffusive predator-prey model with Allee effect in predators. For the system without nonlocal delay, the stability and Turing instability conditions are derived. For the system with nonlocal delay, all Turing bifurcation curves on $ (d_{2}, \tau) $ plane are obtained, and the critical Turing bifurcation curve near the stable region is determined by the discussion of the properties of Turing bifurcation curves. It is found that the presence of nonlocal delay decreases the range of the occurrence for Turing bifurcation. The critical Hopf bifurcation curve is found by the geometric stability switch criterion. The stable region for the positive equilibrium is bounded by the critical Turing and Hopf bifurcation curves. On the boundaries of stable region, Hopf bifurcation, Turing bifurcation, Turing-Turing bifurcation and Turing-Hopf bifurcation may occur. To reveal the complex dynamic near Turing-Hopf bifurcation point, normal form for the system with nonlocal delay is derived, from which different types of bistability are found, including two stable spatially inhomogeneous periodic solutions and two stable spatially inhomogeneous steady states.

In this article, we study Hopf bifurcation and Turing instability of a diffusive predator-prey model with hunting cooperation. For the local model, we analyze the stability of the equilibrium and derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, first it is shown that Turing instability occurs, then the direction and stability of the Hopf bifurcation is reached. Our results show that hunting cooperation plays a crucial role in the dynamics of the model, that is, it can be beneficial to the predator population and induce the rise of Turing instability. Finally, numerical simulations are performed to visualize the complex dynamic behavior.

Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior

The stability of the ecosystem is essential for the sustainable development of the earth. Studying the population model’s dynamic behavior, including the Allee and anxiety effects, can better represent the ecosystem’s working mechanism, which is crucial for preserving ecological balance. In light of this, the objective of this paper is to build a Leslie–Gower model that incorporates Allee effect on the birth rate and the saturated fear effect on the predator, then analyze its dynamic behavior and the impact of the saturated fear effect on population density. In the process of analysis, the existence and stability of boundary and positive equilibria are established, demonstrating that the origin is an attractor using the blow-up method. By varying the saturated fear effect parameter, the corresponding system will undergo supercritical, sub-critical, and even degenerate Hopf bifurcations. The existence of Bogdanov–Takens bifurcation of codimension-2 (or codimension-3) is demonstrated near the unique positive equilibrium. In light of these bifurcation phenomena, the validity of the theoretical results is confirmed through graphical representations via numerical simulations. The results show that while the fear effect on the predator favorably contributes to the ecological stability, high levels of either the Allee effect or the saturated fear effect pose a hazard to the stability of the ecosystem.

Bifurcation analysis in a diffusive predator-prey model with spatial memory of prey, Allee effect and maturation delay of predator

In this paper, we establish a predator–prey model with focus on the Allee effect and prey group defense. The positivity and boundedness of the model, existence of equilibrium point, and stability change caused by Allee effect are studied. Bifurcation (transcritical bifurcation, Hopf bifurcation) analysis is discussed, and the direction of Hopf bifurcation is determined by calculating the first Lyapunov number. Then we introduce delay into the original model and consider the influence of delay on the stability of the model. By selecting delay as the bifurcation parameter, we obtain the existence conditions of Hopf bifurcation and the direction of Hopf bifurcation. Finally, we verify the theoretical analysis by numerical simulation. Considering both the Allee effect and the prey group defense, the dynamic behavior near the origin becomes more complex than only considering Allee effect or prey group defense in the model. Allee effect can bring the risk of extinction and the change of stability, and the delay effect can make the stable coexistence equilibrium unstable and lead to periodic oscillation.

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 [Formula: see text] and [Formula: see text], where [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], 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.

Theories to explain the diversity of species have required that individual species occupy unique niches and/or vary in their response to environmental factors. Positive interactions within a species, although common in communities, have not been thought to maintain species diversity because in non-spatial models the more abundant species always outcompetes the rarer species. Here, we show, using a stochastic spatial model, that positive intraspecific interactions such as those caused by positive frequency dependence and/or priority effects, can maintain species diversity if interactions between individuals are primarily local and the habitat contains areas that cannot be colonized by any species, such as boulders or other physical obstructions. When intraspecific interactions are primarily neutral, species diversity will eventually erode to a single species. When the landscape is homogeneous (i.e. does not contain areas that cannot be colonized by any species), the presence of strong intraspecific interactions will not maintain diversity.

In this paper, the spatiotemporal pattern dynamics of a diffusion population model with Allee effect and environmental toxins is considered. Firstly, the boundedness and positivity of the system are proved, the existence and stability of the equilibrium are analyzed, and the conditions of the occurrence of Turing bifurcation are deduced. Subsequently, the mechanism for the selection of patterns around the positive equilibrium was provided through weakly nonlinear analysis. One finds that weak Allee effect can restore the reduction of population density and undergoes the bistable phenomena. Environmental toxins can lead to Hopf bifurcation and saddle‐node bifurcation, and change the topology of the system. Self‐diffusion can induce Turing bifurcation, and the hexagonal hole patterns, mixed patterns of holes and stripes, and striped patterns are shown. Finally, theoretical results are validated through numerical simulations. These findings provide a scientific basis for population conservation strategies under the threat of environmental toxins.

The reaction-diffusion Holling-Tanner prey-predator model considering the Allee effect on predator, under zero-flux boundary conditions, is discussed. Some properties of the solutions, such as dissipation and persistence, are obtained. Local and global stability of the positive equilibrium and Turing instability are studied. With the help of the numerical simulations, the rich Turing patterns, including holes, stripes, and spots patterns, are obtained.

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.

There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large carrying capacity is assumed) is used in the model.

In this paper, spatiotemporal dynamics of a phytoplankton-fish system with the Allee effect and harvest effect are investigated mathematically and numerically. Mathematical theoretical works have been pursued for the investigation of the stability of the equilibrium point of the phytoplankton-fish system with the Allee effect and harvest effect, which in turn provide a theoretical basic for the numerical simulation. Numerical analysis works indicate that Allee effect and harvest effect have a strong effect on the spatiotemporal dynamics of the phytoplankton-fish system using pattern formation. These results may help us to better understand phytoplankton-fish interactions.

In this paper, a mathematical model is proposed to study a predator–prey system with multiple Allee effect acting on the growth rate of the prey population. Cosner-type functional response is incorporated to model schooling or conspecific aggregation behaviour of predator population. Cosner-type functional response emphasizes the strong Allee effect among the predator’s population. We determine the existence of biological feasible equilibrium points with their stability properties. Besides, bifurcation scenarios such as saddle-node, Hopf and Bogdanov–Takens have been observed that take place in the dynamics during the parameter variation. Allee effect and growth rate of the prey population give rise to the multistable behaviour of the system. We attempt to portrait the dynamical behaviour of the system in the presence of diffusion. It has been observed that the predator adopts different strategies to surround the prey population. These strategies are depending on the prey’s school size, density and predator’s agility. Moreover, severity of the Allee effect also brings dramatic changes in the predator–prey distribution in the system. It is found that under the strong Allee effect, the predator–prey populations form high-density schools which are completely justified in the biological point of view. Various types of patterns for different time steps and diffusion rates are captured that confer the spatiotemporal complexity of the system.