The paper proposes a two-component discrete-time model of the plankton community, taking into account features of the development and interaction of phytoplankton and zooplankton. To describe the interaction between these species and to compare the system dynamics, we use the following set of response functions: type II and III Holling function and the Arditi–Ginzburg response function, each of which describes trophic interactions between phytoplankton and zooplankton. An analytical and numerical study of the model proposed is made. The analysis shows that the variation of trophic functions does not change the dynamic behavior of the model fundamentally. The stability loss of nontrivial fixed point corresponding to the coexistence of phytoplankton and zooplankton can occur through a cascade of period-doubling bifurcations and according to the Neimark–Saker scenario, which allows us to observe the appearance of long-period oscillations representing the alternation of peaks and reduction in the number of species as a result of the predator-prey interaction. As well, the model has multistability areas, where a variation in initial conditions with the unchanged values of all model parameters can result in a shift of the current dynamic mode. Each of the models is shown to demonstrate conditional coexistence when a variation of the current community structure can lead to the extinction of the entire community or its part. Considering the characteristics of the species composition, the model with the type II Holling function seems a more suitable for describing the dynamics of the plankton community. Such a system is consistent with the idea that phytoplankton is a fast variable and predator dynamics is slow; thus, long-period fluctuations occur at high phytoplankton growth rates and low zooplankton ones. The model with the Arditi–Ginzburg functional response demonstrates quasi-periodic fluctuations in a narrow parametric aria with a high predator growth rate and low prey growth rate. The quasi-periodic dynamics regions in the model with the Holling type III functional response correspond to the conception of fast and slow variables, however in this case, the stability of the system increases, and the Neimark-Sacker bifurcation occurs even at a higher growth rate of zooplankton.
The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark-Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark-Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.
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