Modeling the interaction of birds and small fish in a coastal lagoon. (English) Zbl 07533166
Summary: Coastal lagoons are high value productive and important systems for different projects. For example, aquaculture, fisheries and tourism are few of them. The quality of coastal waters in the ecosystems of lagoons can be greatly influenced by the growth of unwanted elements, e.g., excessive fisheries, tourism, etc. In this paper, a mathematical model is proposed and analysed to study the general and simplified form of an ecosystem of Chilika Lake, India. Chilika Lake (19°28’N–19°54’N and 85°06’E–85°36’E) is the largest wintering ground for migrating water fowl found anywhere on the Indian sub-continent. These migratory birds utilize the Chilika Lake for feeding, resting and breeding. The interaction of birds and small fish in the Chilika Lake is considered to be Leslie-Gower Holling type II. Since big fish are being sourced as income for local fishermen and the population of big fish is highly variable, and hence birds and small fishes are mainly the two types of biomass considered for this study. It must be noted that, in this study, we have considered the case of Chilika lake theoretically only and no practical data is collected for this study, and the name of Chilika is used only for better ecological understanding. Therefore, this theoretical study maybe linked to any such ecosystem. Their interaction is found mathematically, a two-dimensional continuous-time dynamical system modeling a simple predator-prey food chain. The dynamical system is represented in the form of two nonlinear coupled ordinary differential equation (ODE) systems. The main mathematical results are given in terms of boundedness of solutions, existence of equilibria, local and global stability of the coexisting interior point. An ecosystem in Indian coastal lagoons may suffer immediate environmental perturbations, such as depressions, tropical cyclones, earthquakes, epidemics, etc. To model such situations, the ODE model is further extended to a stochastic model driven by Lévy noise. The stochastic analysis includes the existence of the unique global solution, stability in mean, and extinction of the population. The proposed model is numerically simulated with the help of an assumed set of parameters for the possible pictorial behavior of the theoretical model. The proposed model may be used for planning purposes by using the data on meteorological and weather shocks such as heavy rainfall, heat-waves, cold-waves, depressions, tropical cyclones, earthquakes, etc. from India Meteorological Department (IMD).
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 70Kxx | Nonlinear dynamics in mechanics |
| 34Cxx | Qualitative theory for ordinary differential equations |
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