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.
Highlights
The study of mathematical models for population dynamics is considered as a key area in abstract ecology from the time when the famous Lotka–Volterra model was presented [1]
8 Concluding remarks We study the dynamics of a discrete-time phytoplankton–zooplankton model [25]
We show that there exist a unique positive fixed point of system (1.5), which is contained in that rectangular region (Theorem 2.2)
Summary
The study of mathematical models for population dynamics is considered as a key area in abstract ecology from the time when the famous Lotka–Volterra model was presented [1]. The authors in [9] have contemplated the effect of predation on competitory elimination and the coexistence of competitory predators They presented and explored a one-phytoplankton two-zooplankton model along with the consideration of harvesting. The functional response αp(t)z(t) a+p(t) represents the predation rate of zooplankton population on phytoplankton species. Phytoplankton population has logistic growth [21] in the absence of zooplankton population, where r is their exponential rate of growth, and k is the maximum carrying capacity of environment Under these conditions we have the following phytoplankton–zooplankton model [22]:. We introduce E as the parameter for combined effort for harvesting of population [25] Under these modifications, system (1.2) takes the following mathematical form: z(t) m1p3(t) q1Ep(t), m2z2(t) q2Ez(t),. We have the following theorem about the boundedness of all solutions of (1.5)
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