A predator-prey model with prey-taxis under homogeneous Dirichlet boundary condi-tions was established.Based on the prior estimate of positive solutions,the asymptotic stability of trivial and semi-trivial solutions was obtained by using the eigenvalue theory of linearization oper-ators and spectral theory.The existence of positive steady-state solutions was established using the fixed point index theory in a cone.By analyzing two related eigenvalues,the coexistence re-gion with respect to the growth rates of two species was obtained.The results show that prey-tax-is has an obvious influence on the steady-state solutions.
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