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

Abstract

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.

A Dynamically Consistent Nonstandard Difference Scheme for a Discrete-Time Immunogenic Tumors Model

“…Using the Euler method or piecewise constant arguments, it is possible to discuss other types of bifurcations and chaos control. We refer readers to [ 44 , 45 , 46 , 47 , 48 ] and the references therein for further consideration. We anticipate that analysis of system ( 3 ) using the Euler method and bifurcation analysis with chaos control for the obtained system will be our future tasks.…”

“…Here, we discuss the Neimark–Sacker bifurcation experienced by system ( 5 ) about under certain mathematical conditions. For further study of bifurcation theory and to better understand this surprising behaviuor of discrete-time mathematical systems, we refer readers to [ 42 , 43 , 44 , 45 , 46 , 47 , 48 ]. Here, we use the standard theory of bifurcation for study of the Neimark–Sacker bifurcation of system ( 5 ) at .…”

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

“…Moreover, our main aim is to study a consistent counterpart of the system (2) such that there is a minimal change in the dynamical behaviour of the discretized system compared to the original continuous system. Therefore, by using Euler's forward method with step size η , we have the following discrete-time version of (2) : Moreover, to obtain a consistent counterpart of system (3) and by applying the Micken's type nonstandard scheme on the model (3) , we get the following discrete-time mathematical model (see [33] ): The next part of this manuscript is structured along these lines: The boundedness character for every positive solution of system (4) is discussed in section 2 . The existence of fixed points and local stability of system (3) and (4) about each of them is investigated in section 3 .…”