Motivated by a prey-predator model with nonautonomous diffusion, we study fully nonautonomous evolution equation of the form \(\frac{d x}{dt} + A(t)x(t) = f(t, x)+g(t)\) in which the family of linear partial differential operators \((A(t))_{t\in \mathbb {R}}\) and the nonlinear function f(t, x) are 1-periodic in time t, whereas the external force g(t) is an almost periodic function. We show the existence of a unique almost periodic solution to the above-mentioned equation. We also prove the existence of an inertial manifold for the solutions around such a solution. The existence of such an inertial manifold is shown in the cases that the family \((A(t))_{t\in \mathbb {R}}\) generates an evolution family \((U(t,s))_{t\ge s}\) satisfying certain dichotomy estimates, and the nonlinear function f(t, x) is \(\varphi \) -Lipschitz, i.e., \(\left||{f(t,x_1)-f(t,x_2)}\right||\leqslant \varphi (t)||{A(t)^{\theta } (x_1-x_2)}||\) where \(\varphi (\cdot )\) belongs to certain admissible space. Then, we apply our abstract results to the prey-predator model with nonautonomous diffusion.
