Infection spreading in tissue as a reaction-diffusion wave.

Viral infection develops in the organism due to virus replication inside infected cells and its transmission from infected to uninfected cells through the extracellular matrix or cell junctions. In this work, we model infection spreading in tissue with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells and virus. We prove the wave existence, determine its speed of propagation and introduce a simplified one-equation model obtained from the complete model using a quasi-stationary approximation.

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.

Influence of Allee effect in prey populations on the dynamics of two-prey-one-predator model.

One of the important ecological challenges is to capture the complex dynamics and understand the underlying regulating ecological factors. Allee effect is one of the important factors in ecology and taking it into account can cause significant changes to the system dynamics. In this work we consider a two prey-one predator model where the growth of both the prey population is subjected to Allee effect, and the predator is generalist as it survives on both the prey populations. We analyze the role of Allee effect on the dynamics of the system, knowing the dynamics of the model without Allee effect. Interestingly we have observed through a comprehensive bifurcation study that incorporation of Allee effect enriches the local as well as the global dynamics of the system. Specially after a certain threshold value of the Allee effect, it has a very significant effect on the chaotic dynamics of the system. In course of the bifurcation analysis we have explored all possible bifurcations such as the existence of transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation and period-doubling route to chaos respectively.

Adaptive Control Based Harvesting Strategy for a Predator-Prey Dynamical System.

This paper deals with designing a harvesting control strategy for a predator-prey dynamical system, with parametric uncertainties and exogenous disturbances. A feedback control law for the harvesting rate of the predator is formulated such that the population dynamics is asymptotically stabilized at a positive operating point, while maintaining a positive, steady state harvesting rate. The hierarchical block strict feedback structure of the dynamics is exploited in designing a backstepping control law, based on Lyapunov theory. In order to account for unknown parameters, an adaptive control strategy has been proposed in which the control law depends on an adaptive variable which tracks the unknown parameter. Further, a switching component has been incorporated to robustify the control performance against bounded disturbances. Proofs have been provided to show that the proposed adaptive control strategy ensures asymptotic stability of the dynamics at a desired operating point, as well as exact parameter learning in the disturbance-free case and learning with bounded error in the disturbance prone case. The dynamics, with uncertainty in the death rate of the predator, subjected to a bounded disturbance has been simulated with the proposed control strategy.

Top-down control in a patchy environment: revisiting the stabilizing role of food-dependent predator dispersal.

In this paper, we revisit the stabilizing role that predator dispersal and aggregation have in the top-down regulation of predator-prey systems in a heterogeneous environment. We consider an environment consisting of sites interconnected by dispersal, and propose a novel mechanism of stabilization for the case with a non-sigmoid functional response of predators. We assume that the carrying capacity of the prey is infinitely large in each site, and show that successful top-down regulation of this otherwise globally unstable system is made possible through an interplay between the unevenness of prey fitness across the sites and the rapid food-dependent migration of predators. We argue that this mechanism of stabilization is different from those previously reported in the literature: in particular, it requires a high degree of synchronicity in local oscillations of species densities across the sites. Prey outbreaks take place synchronously, but the unevenness of prey growth rates across the sites results in a pronounced difference in the species densities, and so the predator quickly disperses to the sites with the highest prey abundances. For this reason, the consumption of prey mostly takes place in the sites with high densities of prey, which assures an efficient suppression of outbreaks. Furthermore, when the total size of prey population is low, the distribution of both species among the sites becomes more even, and this prevents overconsumption of the prey by the predator. Finally, we put forward the hypothesis that this mechanism, when considered in a tri-trophic plankton community in the water column, can explain the stability of the nutrient-rich low-chlorophyll open ocean regions.