Global stability of an age-structured population model on several temporally variable patches

Abstract

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number $$R_0$$ R 0 for time-independent and periodical models and establish the permanence dichotomy: if $$R_0\le 1$$ R 0 ≤ 1 , extinction on all patches is imminent, and if $$R_0>1$$ R 0 > 1 , permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.

On the unique weak solvability of second-order unconditionally stable difference scheme for the system of sine-Gordon equations

“…are used to obtain a priori estimates for the nonnegativity and boundedness of the terms in Eqs. (18), (19) of system (16) (see, e.g., [24]). Next, using the coercivity estimate (9), spectral properties of the operator A, estimates (20), (21), Cauchy-Schwarz inequality, Young's inequality, triangle inequality, and some simple identities together, the following inequalities are obtained:…”

“…The weak solution may have discontinuities or other irregularities, but it still satisfies the problem in a small region. Weak solutions can model the spread of cancer in the body, the dynamics of populations of competing species, and the spread of viruses and other diseases in populations (see [12,18,30]). In mathematical biology, the existence of traveling wave front solution and analysis for Nicholson's blowflies equation are studied in [14] and [15].…”

“…The weak and global solutions, nonlinear dynamics of PDEs, finite-difference and finite-element methods are extensively studied by many scientists (see [1,6,[13][14][15][16][17][18][19][20][21][22][23]28,32] and the references given therein).…”