## Existence of positive almost periodic or ergodic solutions for some neutral nonlinear integral equations.(English)Zbl 1240.42021

The paper investigates the dynamics of a population affected by constant and proportional harvesting. The authors present four models for distributed delay logistic equations with harvesting. Under some conditions these equations are studied by the equations $x'(t)=r x(t) \left (1-\int _0^{t}x(\xi )\, d\xi \right ) - H(x),$
$x'(t)=r x(t) \left (1-\int _0^{t}\alpha e^{-\alpha (t-\xi )} x(\xi )\, d\xi \right ) - H(x),$ where $$H(x)=h$$ or $$H(x)=hx$$ ($$r,h,\alpha$$ are positive constants). The behavior of solutions, the stability of the equilibria and the existence of a Hopf bifurcation are studied. A numerical simulation is performed to illustrate the obtained results.

### MSC:

 42A75 Classical almost periodic functions, mean periodic functions 45G10 Other nonlinear integral equations