Study of a logistic equation with local and non-local reaction terms

Abstract

Abstract. In this work we examine a logistic equation with local and nonlocal reaction terms both for time dependent and steady-state problems. Mainly, we use bifurcation and monotonicity methods to prove the existence of positive solutions for the steady-state equation and sub-supersolution method for the long time behavior for the time dependent problem. The results depend strongly on the size and sign of the parameters on the local and non-local terms.

Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media

“…For more works on (5) with b(x, t) ≡ 0, the reader is referred to [1,25,26], and for more works on nonlocal reaction-diffusion model on the whole space, the reader is referred to [3,4,10,17,18]. For the study of ( 6)-( 7) in time independence case, we refer the reader to [9,13,16,47], in which the authors applied the bifurcation theory and monotonicity methods to study the existence and stability of steady states.…”

Study of a generalized logistic equation with nonlocal reaction term

“…Proof of Theorem 2.1 From Lemmas 2.4 and 2.5 and bifurcation theorem (see [18]), the same proof as that of Theorem 2.2 in [8] guarantees the existence of an unbounded continuum C 0 of positive solutions of (1.1). Moreover, conclusion (i) is true.…”

“…The proof is complete. Conclusion (i) is Proposition 3.1 in [8] and the proof of (ii) is similar to that in Theorem 3.1, and we omit it.…”

“…supercritical) if there exists a neighborhood V of (λ 1 , 0) such that for every solution (λ, u) ∈ V satisfies λ < λ 1 (resp. λ > λ 1 ), see [8].…”

“…In this paper, we consider the nonlocal elliptic boundary value problem This type of problem was studied initially by Delgado et al in [8], where they proposed the equation Here u(x, t) represents the density of a species in time t > 0 and at the point x ∈ , the habitat of the species that is surrounded by inhospitable areas, λ is the growth rate of species, term -f (u) describes the limiting effect of crowding in the population. In this paper the authors proved the existence of an unbounded continuum of positive solutions of (1.3), presented some non-existence results, and discussed the local and global behavior of the continuum.…”