Stationary, non-stationary and invasive patterns for a prey-predator system with additive Allee effect in prey growth

Non-spatial Dynamics and Spatiotemporal Patterns Formation in a Predator-Prey Model with Double Allee and Dome-shaped Response Function

The extinction of species is a major threat to the biodiversity. Allee effects are strongly linked to population extinction vulnerability. Emerging ecological evidence from numerous ecosystems reveals that the Allee effect, which is brought on by two or more processes, can work on a single species concurrently. The cooperative behavior which raises Allee effect in low population density, can create group defence in species to protect themselves from predation. This article focuses on the dynamics of a predator-prey system with double Allee effect in prey growth and simplified Monod-Haldane form of dome-shaped response function to incorporate group defence ability of prey as time and space vary. The study obtains that, to some extent, group defence of prey plays a positive role for the stability of both the species, but on negative side, if defensive ability exceeds a threshold value then both the population can not survive simultaneously and predator population dies out. The Allee effect produces bi-stability (weak Allee) even tri-stability (strong Allee) in phase space reflecting that the system dynamics is very sensitive subject to initial population of the species. The combined impact of double Allee and group defence of prey leads in populations enduring stable periods punctuated by oscillations. The species' mobility based on only its own population is insufficient to this model for Turing instability. The presence of double Allee effect increases the instability regions that enhances the likelihood of various patterns. Whereas increasing group defence of prey decreases the instability region in spatial system. The species distribution stabilizes in forms of spots, stripes and mixture of both in heterogeneous environment. But for prey, gathering decreases with increasing growth rate and gathering increases with increasing Allee effect due to cross-diffusion which results paradox to temporal system. In contrast, populations in the Hopf and Hopf-Turing regions fluctuate (oscillatory) or their distribution becomes unpredictable (chaotic).

Exploring the effects of competition and predation on the success of biological invasion through mathematical modeling

Biological invasions are a major cause of species extinction and biodiversity loss. Exotic predators are the type of introduced species that have the greatest negative impact, causing the extinction of hundreds of native species. Despite this, they continue to be intentionally introduced by humans. Understanding the causes that determine the success of these invasions is a challenge within the field of invasion biology. Mathematical models play a crucial role in understanding and predicting the behavior of exotic species in different ecosystems. This study examines the effect of predation and competition on the invasion success of an exotic generalist predator in a native predator-prey system. Considering that the exotic predator both consumes the native prey and competes with the native predator, it is necessary to study the interplay between predation and competition, as one of these interspecific interactions may either counteract or contribute to the impact of the other on the success of a biological invasion. Through a mathematical model, represented by a system of ordinary differential equations, it is possible to describe four different scenarios upon the arrival of the exotic predator in a native predator-prey system. The conditions for each of these scenarios are described analytically and numerically. The numerical simulations are performed considering the American mink (Mustela vison), an invasive generalist predator. The results highlight the importance of considering the interplay between interspecific interactions for understanding biological invasion success.

The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey-predator model

Pattern formation is a central process that helps to understand the individuals' organizations according to different environmental conditions. This paper investigates a nonlocal spatiotemporal behavior of a prey-predator model with the Allee effect in the prey population and hunting cooperation in the predator population. The nonlocal interaction is considered in the intra-specific prey competition, and we find the analytical conditions for Turing and Hopf bifurcations for local and nonlocal models and the spatial-Hopf bifurcation for the nonlocal model. Different comparisons have been made between the local and nonlocal models through extensive numerical investigation to study the impact of nonlocal interaction. In particular, a legitimate range of nonlocal interaction coefficients causes the occurrence of spatial-Hopf bifurcation, which is the emergence of periodic patterns in both time and space from homogeneous periodic solutions. With an increase in the range of nonlocal interaction, the whole Turing pattern suppresses after a certain threshold, and no pure Turing pattern exists for such cases. Specifically, at low diffusion rates for the predators, nonlocal interaction in the prey population leads to the extinction of predators. As the diffusion rate of predators increases, impulsive wave solutions emerge in both prey and predator populations in a one-dimensional spatial domain. This study also includes the effect of nonlocal interaction on the invasion of populations in a two-dimensional spatial domain, and the nonlocal model produces a patchy structure behind the invasion where the local model predicts only the homogeneous structure for such cases.

Turing Instability and Spatiotemporal Pattern Formation Induced by Nonlinear Reaction Cross-Diffusion in a Predator–Prey System with Allee Effect

The Allee effect is widespread among endangered plants and animals in ecosystems, suggesting that a minimum population density or size is necessary for population survival. This paper investigates the stability and pattern formation of a predator–prey model with nonlinear reactive cross-diffusion under Neumann boundary conditions, which introduces the Allee effect. Firstly, the ODE system is asymptotically stable for its positive equilibrium solution. In a reaction system with self-diffusion, the Allee effect can destabilize the system. Then, in a reaction system with cross-diffusion, through a linear stability analysis, the cross-diffusion coefficient is used as a bifurcation parameter, and instability conditions driven by the cross-diffusion are obtained. Furthermore, we show that the system (5) has at least one inhomogeneous stationary solution. Finally, our theoretical results are illustrated with numerical simulations.

Pattern Formation in a Three-Species Cyclic Competition Model

In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka-Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.

Invasive dynamics for a predator-prey system with Allee effect in both populations and a special emphasis on predator mortality

We considered a non-linear predator-prey model with an Allee effect on both populations on a two spatial dimension reaction-diffusion setup. Special importance to predator mortality was given as it may be often controlled through human-made harvesting processes. The local dynamics of the model was studied through boundedness, equilibrium, and stability analysis. An extensive numerical stability analysis was performed and found that bi-stability is not possible for the non-spatial model. By analyzing the spatial model, we found the condition for successful invasion and the persistence region of the species based on the predator Allee effect and its mortality parameter. Four different dynamics in this region of the parameter space are mainly explored. First, the Allee effect on both populations leads to various new types of species spread. Second, for a high value of per-capita growth rate, two completely new spreads (e.g., sun surface, colonial) have been found depending on the Allee effect parameter. Third, the Allee coefficient on the predator population leads to spatiotemporal chaos via a patchy spread for both linear and quadratic mortality rates. Finally, a more rigorous analysis is performed to study the chaotic nature of the system within the whole persistence domain. We have studied the possibility of chaos through temporal variation in different invasion regions. Furthermore, the chaotic fluctuation is studied through the sensitivity of initial conditions and by investigating the dominant Lyapunov exponent value.

Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

Dynamics of diffusive modified Previte-Hoffman food web model

This paper formulates and analyzes a modified Previte-Hoffman food web with mixed functional responses. We investigate the existence, uniqueness, positivity and boundedness of the proposed model's solutions. The asymptotic local and global stability of the steady states are discussed. Analytical study of the proposed model reveals that it can undergo supercritical Hopf bifurcation. Furthermore, analysis of Turing instability in spatiotemporal version of the model is carried out where regions of pattern creation in parameters space are obtained. Using detailed numerical simulations for the diffusive and non-diffusive cases, the theoretical findings are verified for distinct sets of parameters.

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.