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

Instability of oscillations in the Rosenzweig-MacArthur model of one consumer and two resources

The system of two resources _R and _R and one consumer C is investigated within the Rosenzweig-MacArthur model with a Holling type II functional response. The rates of consumption of particular resources are normalized as to keep their sum constant. Dynamic switching is introduced as to increase the variable C in a process of finite speed. The space of parameters where both resources coexist is explored numerically. The results indicate that oscillations of C and mutually synchronized _R, which appear equal for the rates of consumption, are destabilized when these rates are modified. Then, the system is driven to one of fixed points or to a limit cycle with a much smaller amplitude. As a consequence of symmetry between the resources, the consumer cannot change the preferred resource once it is chosen.

Predator–Prey Models: A Review of Some Recent Advances

In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles.

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.

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.