Moitri Sen

Bifurcation analysis of a ratio-dependent prey–predator model with the Allee effect

A B S T R A C T There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee effect. We both consider the cases of a strong Allee effect (the population growth rate is negative at low species density) and the case of a weak Allee effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee effect results in a paradox of biological control.

Global dynamics of an additional food provided predator–prey system with constant harvest in predators

Abstract The article aims to study the global dynamics associated with a predator prey system when the predator is provided with additional food and harvested at a constant rate. This study supplements the existing literature on the dynamics of additional food provided predator prey system by focusing on the consequences of harvesting the predators. It presents a comprehensive view on the entire range of bifurcations that take place in the considered system and highlights the dependence of the system dynamics on its vital parameters. This study provides important tools for investigations pertaining to controllability of the system which are essential from the real world applications perspective.

Rich Global Dynamics in a Prey–Predator Model with Allee Effect and Density Dependent Death Rate of Predator

In this work we have considered a prey–predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov–Takens bifurcation and Bautin bifurcation respectively.

Pattern Formation in a Prey-Predator Model with Nonlocal Interaction Terms

We study a spatio-temporal prey-predator model with nonlocal interaction terms. Nonlocal interactions are considered for prey and predator species to describe the nonlocal intra-specific competition for limited resources. We show that the region of pattern formation increases with the increase of the range of nonlocal interaction. Numerical continuation technique is used to determine the existence of multiple stationary patterns.

A model for which toxic and non-toxic phytoplankton are indistinguishable by the zooplantkon

Using Haldane-Monod functions, we study the response of zooplankton avoiding to feed on phytoplankton when the latter releases toxines, but assuming that the zooplankton does not distinguish between toxic and non-toxic phytoplankton.

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.

Original article: A model for biological control in agriculture

Persistence and global stability of the coexistence equilibrium of a recently published model in biocontrol of crops are here shown both in the absence and the presence of delays, introduced to simulate the handling time of the prey. In the latter case, the system can behave in two different ways, in dependence of whether a suitably defined key parameter exceeds a certain threshold. Namely, below the threshold the delay is shown not to be able to influence the stability of the coexistence equilibrium; above it, existence of a Hopf bifurcation is analytically proven. Further, in this range, numerical simulations reveal a route to chaotic behavior as function of the size of the delay. Some operative conclusions for agroecosystem management are drawn, although they ultimately depend on each particular situation.

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.

Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect

A B S T R A C T There is a growing body of evidence supporting implementation of ratio-dependent functional response of predators in ecological models. Those models often provide a satisfactory explanation of the observed patterns of dynamics which cannot be done based on the ‘classical’ models using the prey-dependent functional response. Surprisingly enough, all theoretical analysis of ratio-dependant predator–prey interactions has so far been completed only for the simplest case where the prey growth is logistic. In a large number of ecologically relevant situations, however, the growth rate of a population is subject to an Allee effect and the per capita growth rate increases with population density. Taking into account Allee dynamics for the prey growth in models can alter the previous theoretical findings obtained for the logistic growth paradigm. In this paper, we analyse a ratio-dependent predator–prey system with prey growth subject to an Allee effect. We both consider the cases of a strong Allee effect (the population growth rate is negative at low species density) and the case of a weak Allee effect (the population growth is positive at low population density). For both cases we fulfil a comprehensive bifurcation analysis, constructing the parametric diagrams, and also show possible phase portraits. Then we compare the properties of the ratio-dependant predator–prey model with and without the Allee effect and show a substantial difference in the dynamical behaviour of those systems. We show that including an Allee effect in a ratio-dependent predator–prey model removes the possibility of sustainable oscillations of species densities (population cycles). We show that the ratio-dependent predator–prey model with the Allee effect can solve the paradox of enrichment. However, unlike the same model with logistic growth, incorporating the Allee effect results in a paradox of biological control.