This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value $\lambda_1^D(\Omega_0)$, and demonstrate that the existence of the predator in $\overline{\Omega}_0$ only depends on the relationship of the growth rate $\mu$ of the predator and $\lambda_1^D(\Omega_0)$, not on the prey. Furthermore, when $\mu<\lambda_1^D(\Omega_0)$, we obtain the existence and uniqueness of its positive steady state solution, while when $\mu\geq \lambda_1^D(\Omega_0)$, the predator and the prey cannot coexist in $\overline{\Omega}_0$. Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $\overline{\Omega}_0$, which is different from that of the classical Lotka-Volterra predator-prey model.
Xianzhong ZENG
,
Lingyu LIU
,
Weiyuan XIE
. EXISTENCE AND UNIQUENESS OF THE POSITIVE STEADY STATE SOLUTION FOR A LOTKA-VOLTERRA PREDATOR-PREY MODEL WITH A CROWDING TERM[J]. Acta mathematica scientia, Series B, 2020
, 40(6)
: 1961
-1980
.
DOI: 10.1007/s10473-020-0622-7
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