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

Ait Dads, E.; Cieutat, P.; Lhachimi, L.

Differ. Integral Equ. 22, No. 11-12, 1075-1096 (2009).

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.

Reviewer: Svatoslav Staněk (Olomouc)

MSC:

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

Keywords:

neutral integral equation; periodic solution; almost periodic solution; asymptotically almost periodic solution; Hilbert’s projective metric