Mathematical Model of ALLEE Effect on Animal Single Population Dynamics and Behavior of Solutions
The rapid decline of species diversity is one of the most serious problems facing the world today, but habitat fragmentation caused by human activities is a major cause of loss of species diversity at present. Habitat fragmentation can lead to habitat loss and patch separation. Fragmentation under natural conditions is usually accompanied by habitat loss, therefore, the effect of fragmentation may be confused, and the effect of fragmentation is not yet clear. In addition, fragmentation often leads to the emergence of small populations; small populations are more susceptible to Alice effect, which may lead to negative population growth or eventual extinction. It has a great influence on the survival of population and the interaction between species. So when studying fragmentation, considering the role of Alice effect not only helps to better understand the mechanism of habitat fragmentation, it is also important for species diversity conservation. In this paper, Alice effect is introduced into population growth, interspecific competition and predator-prey model respectively, a fragmentation process is simulated: a complete habitat patch is fragmented into two small patches. The sum of environmental carrying capacity of the latter is equal to that of the larger patches. According to the characteristics of single population dynamic ALLEE effect, an appropriate mathematical model is introduced, the oscillation, boundedness and asymptotic stability of the solution are studied. It is found that the stronger the ALLEE effect, the worse the stability of the system, further reduce the possibility of species coexistence.