The authors consider and derive stage structured models with immature and mature individuals, assuming that \(\tau>0\) is a fixed threshold age. The key point in this article is that the intra- and inter-specific competitions at immature life stage (larval stage) are considered.
For a single species (let \(u(t,a)\) be the age density at time \(t\) and age \(a\)), there are two modelling ways using different birth laws. One is \(u(t,0)=b(A(t)),\) where \(A(t):=\int_{\tau}^{\infty}u(t,a)da \;(t\geq 0)\) is the total number of sexually mature adults. Another is \(u(t,0)=N(t)=\int_{\tau}^{\infty}b(a)u(t,a)da \;(t\geq 0)\). The former leads to a prototype delay equation, the solution \(A(t)\) of which is bounded for any egg laying rate \(b(\cdot)\). It also generates a monotone dynamical system if \(b(\cdot)\) is monotone increasing. The latter yields a nonlinear integral equation, the solution \(N(t)\) of which is also bounded under some further assumptions.
For two species with larval intra- and inter-specific competition, “a system of delay equations cannot be always be written down explicitly because their right hand sides depend on the solutions of a nonlinear ODE system”, and thus it is difficult to make progress with the further study. However, by using some known results in [Y. Kuang, Delay differential equations: with applications in population dynamics. Mathematics in Science and Engineering. 191. Boston, MA: Academic Press, Inc. (1993; Zbl 0777.34002); S. B. Hsu et al., Trans. Am. Math. Soc. 348, No. 10, 4083–4094 (1996; Zbl 0860.47033)], some linear stability about any boundary equilibrium \((U_*,0)\) or \((0,V_*)\) could be obtained.