Persistence, contractivity and global stability in logistic equations with piecewise constant delays

By Y. Muroya. 2002

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Abstract We establish sufficient conditions for the persistence and the contractivity of solutions and the global asymptotic stability for the positive equilibrium N ∗ =1/(a+∑ i=0 m b i ) of the following differential equation with piecewise constant arguments: dN(t) dt =N(t)r(t) 1−aN(t)− ∑ i=0 m b i N(n−i) , n⩽t n=0,1,2,…, N(0)=N 0 >0 and N(−j)=N −j ⩾0, j=1,2,…,m, where r(t) is a nonnegative continuous function on [0,+∞), r(t)≢0, ∑i=0mbi>0, bi⩾0, i=0,1,2,…,m, and a+∑i=0mbi>0. These new conditions depend on a,b0 and ∑i=1mbi, and hence these are other type conditions than those given by So and Yu (Hokkaido Math. J. 24 (1995) 269–286) and others. In particular, in the case m=0 and r(t)≡r>0, we offer necessary and sufficient conditions for the persistence and contractivity of solutions. We also investigate the following differential equation with nonlinear delay terms: dx(t) dt =x(t)r(t){1−ax(t)−g(x([t]),x([t−1]),…,x([t−m]))}, t⩾0, x(−k)=φ(−k)⩾0, 0⩽k⩽m, and φ(0)>0, where r(t) is a nonnegative continuous function on [0,+∞), r(t)≢0, 1−ax−g(x,x,…,x)=0 has a unique solution x ∗ >0 and g(x0,x1,…,xm)∈C1[(0,+∞)×(0,+∞)×⋯×(0,+∞)].

Subfields: Applied Mathematics, Numerical Analysis, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/85478fc6559f907e0e70f7a1246c9f7b33ebdbbb

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