Role of diffusion strategies and harvesting in competition for spatially distributed populations in a heterogeneous environment

Abstract

The loss and degradation of habitat, Allee effects, climate change, deforestation, hunting-overfishing, and human disturbances are alarming and significant threats to the extinction of many species in ecology. In this study, the reaction-diffusion equations on the competition outcome that describe the dynamics of single species and two competing species in a heterogeneous environment are accomplished. The main goal of this research is to study the influence of different diffusion strategies with different growth laws where the harvesting effects are carried out under various conditions. As it is known that, if the resource distribution significantly varies, the standard diffusion model does not always appear realistic, as in some situations, there is a migration of species from resourceful to resource poor locations. An alternative type of diffusion strategy known as resource-based diffusion strategy has been considered in this research where rather than the population itself, its ratio to either locally available resources (carrying capacity) or to a positive distribution function diffuses. The study concentrated on how the selection is influenced by resource-based diffusion, particularly carrying capacity-driven dispersion in the environment. For the single species population model Gilpin-Ayala type growth with harvesting has been considered and determine conditions on harvesting rate for persistence and extinction of species based on the global stability of the non-trivial and trivial equilibrium states of the solution. The numerical results are presented on the nature of nonzero equilibrium states and their dependence on resource function, capacity functions, migration coefficient as well as on Gilpin-Ayala parameter. For two competing species, a clear conception of the stability (competitive exclusion and coexistence) of the competing species, their growth and extinction have been demonstrated in the research. Two species competition models for generalized symmetric growth laws with different types of dispersal strategies: one of them is subject to a carrying capacity-driven diffusion while the other follows the resource-based diffusion strategy is studied. If spatially heterogeneous dispersion functions are non-proportional, then competitive exclusion of a resource-based diffusion population is inevitable, and coexistence is not possible unless the whole environment is homogeneous. However, in the case of proportionality, the species shows similar behavior for the different non-constant and unequal carrying capacities of the competing species. The coexistence is possible for different resource consumption if the carrying capacity and the resource function of the second organism are identical and non-constant. On the other hand, when both species follow the resource-based diffusion strategy with the same logistic growth law, but the migration coefficient differs, then based on different relationships between the resource function and carrying capacity the global coexistence and competitive exclusion will occur in competition. Additionally, it is found that a higher migration rate has a negative impact to sustain and reduce the population density in competition. However, the study explored the influence of harvesting on the competition outcome in the two species competition model under different conditions and situations. Also, find the condition for coexistence and extinction of species based on the relation between different harvesting levels and intrinsic growth rate. Some estimates on harvesting bounds for which coexistence solution should exist are provided. For non-symmetric growth, the case of the ideal free pair as a combination of the two strategies adopted by the two species is considered and find that the relevant coexistence equilibrium is a global attractor. Based on the values of spatially dependent competition coefficients and different imposed diffusion strategies the case of weak and strong competition are studied. In case of weak competition, both species can coexist even if only one of the diffusion strategies is proportional to the carrying capacity. For strong competition, coexistence is not attainable, and find the competitive exclusion of the other; the populations are locally survive. Additionally, to see the evolution of dispersal, a new type of reaction-advection-diffusion model for a large group of growth laws is considered and analyzed the global stability of coexistence and competitive exclusion of the species based on different conditions on model parameters. Moreover, for better understanding, all the considered models are studied numerically both for one and two space dimensions for time-dependent and independent cases, which are extremally new in this study. As the theory does not give any idea about the shape of the non-zero equilibrium state, our study explores these numerically. Since it is very essential for ecological considerations.