Debaldev Jana

Chaotic dynamics of a discrete predator-prey system with prey refuge

Prey-predator system with prey refuge for the case of non-overlapping generations is considered here for our present study. Foraging efficiency of predator largely varies with the escaping ability of the prey from the predator. For this reason system experiences interesting and complex dynamical features with varying the strength of prey refuge, including both population extinction, predator extinction, stable coexistence, multiple invariant closed orbit in different chaotic regions, onset of chaos suddenly and sudden disappearance of the chaotic dynamics. In particular, we observe that when the prey is in stable, oscillatory or even chaotic status, then the predator can tend to extinction. Also, system experiences Hopf-bifurcation and flip bifurcations. Numerical computation is also performed to validate and visualize different theoretical results. The computations of Lyapunov exponent, fractal dimension of the map, recurrence plot and power spectral density confirm the chaotic dynamical behaviors. The analysis and results in this paper are interesting both in mathematics and biology.

Dynamics of generalist predator in a stochastic environment

Modified ratio-dependent Holling-Tanner model with prey refuge is considered.Delays are taken in logistic growth of prey as feedback mechanism and gestation period of predator.Model system exhibits Hopf-bifurcation when the delay parameters cross their critical values.Stochastic analysis of the model system is carried out by incorporating Gaussian white noise.Using Fourier transform technique, fluctuation and stability of the stochastic model is studied. In this paper, an attempt has been made to understand the dynamics of a prey-predator system with multiple time delays where the predator population is regarded as a generalist type. In this regard, we consider a modified Holling-Tanner prey-predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predators interference in prey-predator relationship provides better descriptions of predators feeding over a range of prey-predator abundances, so the predators functional response is considered to be Type II ratio-dependent and foraging efficiency of predator largely varies with the refuge strategy of prey population. In accordance with previous studies, it is observed that delay destabilizes the system, in general and stability loss occurs via Hopf-bifurcation. In particular, we show that there exists critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. Also, environmental stochasticity in the form of Gaussian white-noise plays a significant role to describe the system and its values. Numerical computation is also performed to validate and visualize different theoretical results presented. The analysis and results in this work are interesting both in mathematical and biological point of views.

Stabilizing Effect of Prey Refuge and Predator’s Interference on the Dynamics of Prey with Delayed Growth and Generalist Predator with Delayed Gestation

In the present paper, I study a prey-predator model with multiple time delays where the predator population is regarded as generalist. For this regard, I consider a Holling-Tanner prey-predator system where a constant time delay is incorporated in the logistic growth of the prey to represent a delayed density dependent feedback mechanism and the second time delay is considered to account for the length of the gestation period of the predator. Predator’s interference in predator-prey relationship provides better descriptions of predators feeding over a range of prey-predator abundances, so the predators functional response here is considered to be Type II ratio-dependent. In accordance with previous studies, it is observed that delay destabilizes the system, in general, and stability loss occurs via Hopf bifurcation. There exist critical values of delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when delay parameters cross their critical values. When delay parameters are large enough than their critical values, the system exhibits chaotic behavior and this abnormal behavior may be controlled by refuge. Numerical computation is also performed to validate different theoretical results. Lyapunov exponent, recurrence plot, and power spectral density confirm the chaotic dynamical behaviors.

Effects of animal dispersal on harvesting with protected areas

Effects of density dependent as well as independent dispersal modes between a harvested patch and a protected area on the maximum sustainable yield and population abundance are studied. Without dispersal, the Gordon-Schaefer harvesting model predicts that as the protected area increases, population abundance increases too but the maximum sustainable yield (MSY) decreases. This article shows that dispersal can change this prediction. For density independent balanced and fast dispersal, neither the MSY, nor population abundance depends on the protected area. For fast and unbalanced dispersal both the MSY and equilibrium population abundance are unimodal functions of the protected area size. For density dependent dispersal which is in direction of increasing fitness predictions depend on whether individuals react to mortality risk in harvested patch. When animals disregard harvesting risk, the results are similar to the case of density independent and balanced dispersal. When animals do consider harvesting risk, the results are similar to the case without dispersal. The models considered also show that dispersal reduces beneficial effect of protected areas, because population abundance is smaller when compared with no dispersal case.

Impact of physical and behavioral prey refuge on the stability and bifurcation of Gause type Filippov prey-predator system

After the pioneering theoretical studies of Lotka and Volterra, Gause and his co-workers replace the previously used linear functional response by using a saturating functional response with a discontinuity at a threshold prey density. Here we assume that prey density at below this threshold value is effectively and successfully in a refuge patch. In this situation there is no food option for predator and go to extinction. But above this threshold value surplus density of prey is available to predator for its diet. But the system does not show any future activities when prey density is in the vicinity of the threshold density, system is ill posed because the trajectories are not well defined here. In the present study, we redefine and analyze the model by using Filippov regularization method. By this continuation method, the system becomes well posed and gives more results as predicted by Gause. Also predator fully depends upon alternative diet to survive from extinction risk when prey is in refuge patch and system largely varies with the availability of alternative diet resource but in the later case predator again switches to its primary (essential) food. When prey density is in the vicinity of the threshold density, then predator may choose its deit preferentially from essential or alternative resources according to its profit. Numerical examples support these hypothesis and analytical results.

Age-structured predator–prey model with habitat complexity: oscillations and control

In this article, we study a predator–prey interaction in a homogeneously complex habitat where predator takes a fixed time to develop from immature to its mature stage. The age-structure of the predator and its interaction with the prey is framed in a system of delay differential equations. The objective is to study the role of habitat complexity and the maturation delay of the predator on the overall dynamics of the model system. Different interesting dynamical behaviours can be obtained by regulating two key parameters, namely the degree of habitat complexity and the maturation delay. It is observed that the system becomes unstable from its stable condition when the maturation delay crosses some critical value. The periodic solutions bifurcated from the interior equilibrium is found to be supercritical and stable. Synchronization of population fluctuations is, however, possible by increasing the strength of habitat complexity. The predator population goes to extinction and the prey population reaches to its maximum, irrespective of the length of maturation delay, when the habitat complexity crosses some upper critical value. The qualitative dynamical behaviours of the model system are verified with the data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction.

Complex dynamics generated by negative and positive feedback delays of a prey–predator system with prey refuge: Hopf bifurcation to Chaos

Various field and laboratory experiments show that prey refuge plays a significant role in the stability of prey–predator dynamics. On the other hand, theoretical studies show that delayed system exhibits a much more realistic dynamics than its non-delayed counterpart. In this paper, we study a multi-delayed prey–predator model with prey refuge. We consider modified Holling Type II response function that incorporates the effect of prey refuge and then introduce two discrete delays in the model system. A negative feedback delay is considered in the logistic prey growth rate to represent density dependent feedback mechanism and a positive feedback delay is considered to represent the gestation time of the predator. Our study reveals that the system exhibits different dynamical behaviors, viz., stable coexistence, periodic coexistence or chaos depending on the values of the delay parameters and degree of prey refuge. The interplay between two delays for a fixed value of prey refuge has also been determined. It is noticed that these delays work in a complementary fashion. In addition, using the normal form theory and center manifold argument, we derive the explicit formulae for determining the direction of the bifurcation, the stability and other properties of the bifurcating periodic solutions.

Interplay between strong Allee effect, harvesting and hydra effect of a single population discrete-time system

The dynamics of a single population with non-overlapping generations can be described deterministically by a scalar difference equation in this study. A discrete-time Beverton–Holt stock recruitment model with Allee effect, harvesting and hydra effect is proposed and studied. Model with strong Allee effect results from incorporating mate limitation in the Beverton–Holt model. We show that these simple models exhibit some interesting (and sometimes unexpected) phenomena such as the hydra effect, sudden collapses and essential extinction. Along with this, harvesting is a socio-economic issue to continue any system generation after generation. Different dynamical behaviors for these situations have been illustrated numerically also. The biological implications of our analytical and numerical findings are discussed critically.

Interaction between prey and mutually interfering predator in prey reserve habitat: Pattern formation and the Turing–Hopf bifurcation

Abstract In this paper, we propose a diffusive prey-predator system with mutually interfering predator (Crowley–Martin functional response) and prey reserve. In particular, we develop and analyze both spatially homogeneous model based on ordinary differential equations and reaction-diffusion model. We mainly investigate the global existence and boundedness of positive solution, stability properties of homogeneous steady state, non-existence of non-constant positive steady state, conditions for Turing instability and Hopf bifurcation of the diffusive system analytically. Conventional stability properties of the non-spatial counterpart of the system are also studied. The analysis ensures that the prey reserve leaves stabilizing effect on the stability of temporal system. The prey reserve and diffusive parameters leave parallel impact on the stability of the spatio-temporal system. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits diffusion controlled pattern formation by different interesting patterns.

On the stability and Hopf-bifurcation of a multi-delayed competitive population system affected by toxic substances with imprecise biological parameters

In this paper we have analyzed the stability and Hopf-bifurcation behaviors of a multi-delayed two-species competitive system affected by toxic substances with imprecise biological parameters. We have exercised a method to handle these imprecise biological parameters by using parametric form of interval numbers. We have studied the feasibility of various equilibrium points and their stability. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation due to variation in the delay parameters which lead to Hopf-bifurcation. Numerical simulations with a hypothetical set of data have been done to support the analytical findings.