Holling type-II Response Research Articles

Modeling the herd prey response to individualistic predators attacks

In this paper, we consider predators hunting on prey gathered in groups and in such way exhibiting the possibility of reducing the predators pressure. To model this feature, however, we depart from the Holling type II (HTII) response function, in that we assume that a sufficiently large set of prey could respond to individualistic attacks and therefore induce the predator to renounce. The basic idea is described at first in a simple two‐populations predator‐prey system. It is then expanded considering the generalist predators to deal with two prey. In the first case, both are gathered in herds, and in the second one, one of the two instead behaves individualistically. The net outcome is an enhanced survival for the prey with respect to both the herding cases without and with predators feeding satiation (i.e., using the HTII response).

On the qualitative study of a discrete-time phytoplankton-zooplankton model under the effects of external toxicity in phytoplankton population

The current manuscript studies a discrete-time phytoplankton-zooplankton model with Holling type-II response. The original model is modified by considering the condition that the phytoplankton population is getting infected with an external toxic substance. To obtain the discrete counterpart from a continuous-time system, Euler's forward method is applied. Moreover, a consistent discrete-time phytoplankton-zooplankton model is obtained by using a nonstandard difference scheme. The boundedness character for every positive solution is discussed, and the local stability of obtained system about each of its fixed points is discussed. The existence of period-doubling bifurcation at a positive equilibrium point is discussed for the discrete system obtained by Euler's forward method. In addition, the comparison of the consistent discrete-time version with its inconsistent counterpart is provided. It is proved that the discrete-time system obtained by using a nonstandard scheme is dynamically consistent as there is no chance for the existence of period-doubling bifurcation in that system. In order to control the period-doubling bifurcation and Neimark-Sacker bifurcation, an improved hybrid control strategy is applied. Finally, we have provided some interesting numerical examples to explain our theoretical results.

Bifurcation and Chaos in a Phytoplankton–Zooplankton Model with Holling Type-II Response and Toxicity

Phytoplankton and zooplankton’s interconnection coordinate in various dynamical processes that occur in ecological population and make it a fascinating subject matter to explore. Time delay is an additional factor that plays an imperative part in the dynamical frameworks. For a stable aquatic ecosystem, the growth of both zooplankton and phytoplankton should be in steady state but in previous eras, there have been a universal explosion in destructive plankton or algal blooms. Many investigators used various mathematical methodologies to try to explain the bloom phenomenon. So in this paper, we studied dynamical characteristics of a discrete-time phytoplankton–zooplankton model with Holling type-II predational functional response and toxic substance distribution. More specifically, we studied local stability with topological classification at equilibrium solutions, periodic points and possible bifurcation analysis of the model under consideration. It is proved that at trivial and semitrivial equilibrium solutions the phytoplankton–zooplankton model does not undergo flip bifurcation but it undergoes Neimark–Sacker bifurcation at interior equilibrium solution. It is also proved that the interior equilibrium solution model does not undergo the fold bifurcation. We also studied chaos in the sense of feedback control strategy. Finally, extensive numerical simulations are provided to demonstrate theoretical results.

Qualitative analysis of a discrete-time phytoplankton\u2013zooplankton model with Holling type-II response and toxicity

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.