Abstract
A ratio-dependent diffusion predator-prey system with free boundary is investigated to understand the impact of free boundary on spreading-vanishing dichotomy and a long time behavior of species. The existence and uniqueness of solutions are verified and the behavior of positive solutions is considered for this system. Moreover, the criteria for spreading-vanishing dichotomy are also derived. The results show that if the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, then the spreading of predators always happens under relatively small rate of death for the predator. On the other hand, it is found that if the initial value of free boundary is smaller than a threshold value and the moving coefficient of free boundary is relatively small depending on initial size of predator or the rate of death is relatively big, the predators fail in spreading to new environment.
Highlights
In mathematical ecology, the invasion of immigration for the new species is one of the most important topics
In [ ], the free boundary model is proposed with a logistic diffusion equation:
If the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, the spreading of predators always happens under relatively small rate of death for the predator
Summary
The invasion of immigration for the new species is one of the most important topics. The behavior of predating always changes by the change of the size of preys and many ecologists observe that the ratio-dependent functional response is more reasonable to describe the process of predating for some predators Based on these facts, we consider the following ratiodependent reaction-diffusion predator-prey system with free boundary including a death. If the length of the initial occupying area is longer than a critical size for the predators or the length of the initial occupying area is shorter than a critical size, but the moving coefficient of free boundary is relatively big, the spreading of predators always happens under relatively small rate of death for the predator. ); ( ) new comparison principle is established and it is used to investigate the criteria for spreading and vanishing; ( ) one initial occupying critical size h is found It follows from the Stefan condition that h (t) > for t ∈ ( , T )
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