EQUATIONS FOR A THREE-STAGED POPULATION MODEL WITH AGE-SIZE-STRUCTURE

In this paper, we present an egg-juvenile-adult model, in which eggs and juveniles are structured by age, while adults by size. The model consists of three first-order partial differential equations with initial and boundary conditions. We derive sensitivity equations for the solutions with respect to the fecundity and mortality of adults.

Sensitivity analysis has been a long-standing and distinguished tool in population ecology.It asks the question what is the linear response of some variable of interest to a change in some parameter.The derivation of sensitivity equations for population models (especially for matrix models) has drawn the attention of numerous researchers in the past few decades because the resulting sensitivity functions can be used in many areas, such as optimization and design [26][27][28], computation of standard errors [29][30].Sensitivity functions can also be used in information theory, control theory, parameter estimation and inverse problems [29][30][32][33][34][35][36].However, little work has been done on the derivation of sensitivity equations for continuous structured population models.
To the best of our knowledge, the work of Banks, Ernstberger and Hu in [24] is the first literature on sensitivity equations and related analysis for size-structured population models, they considered the classical Sinko-Streifer size-structured population model and derived partial differential equations for the sensitivities of solutions with respect to initial conditions, growth rate, mortality and fecundity.Sample numerical results to illustrate use of these equations were also presented.In [25], Ackleh, Deng and Yang examined a model describing the dynamics of an amphibian population whose individuals were divided into juveniles and adults.They made sensitivity analysis for the solutions to the reproduction and mortality of adults.
Since there are many species whose life history consists of more than two stages (e.g.frogs and invertebrates), in this paper we propose an egg-juvenile-adult model.In the first two stages we consider age difference while size in the last one.The remainder of the paper is organized as follows.In Section 2, we present the model and state some assumptions.The existence result for directional derivatives with respect to parameters are established in Section 3. Section 4 derives sensitivity equations for the solutions with respect to reproduction and mortality.The final section contains some remarks.

The model description
In this paper, We propose the following three-staged population model with age-size-structure: ) ) ) ) ) where n e (x,t), n j (x,t) and n a (s,t) denote the densities of eggs and juveniles of age x and adults of size s, respectively, at time t.The parameters µ e , µ j and µ a are mortality for eggs, juveniles and adults, respectively.The function γ(x) is the rate an egg of age x becomes juvenile.The functions β and g are the fertility and growth rates of adults, respectively.x 1 and x 2 denote the maximum age of eggs and juveniles, respectively.Equation (2.6) says that x 2 is the age at which juveniles mature into adults of minimum size s 1 , and s 2 denotes the maximum size of adults.
We first introduce the definition of the solution of problem (2.1)-(2.9)via the method of characteristics.
Definition 2.1 A triple of integrable nonnegative functions (n e (x,t), n j (x,t), n a (s,t)) is said to be a solution of system (2.1)-(2.9)if it satisfies the following equations: )n e (x,t), with where Φ(t; s 0 ,t 0 ) is the solution of the differential equation s (t) = g(s(t)) with initial condition By (A4), the function Φ is strictly increasing, and therefore a unique inverse function Γ(s; s 0 ,t 0 ) exists.Let Z(s) = Γ(s; s 1 , 0); then (s, Z(s)) represents the characteristic curve passing through (s 1 , 0), and this curve divides the (s,t)-plane into two parts.

Directional derivatives with respect to parameters
The main goal of this section is to prove the existence of the directional derivatives with respect to parameters, which will be needed in the derivation of the sensitivity equations.To do so, we introduce the directional derivative of a function f with respect to a parameter θ (see [24]): Definition 3.1 Let Θ be a convex subset in some topological vector space, and f : R + × Θ → R.
Given θ and ϑ in Θ, we define the derivative f θ (t; θ , ϑ − θ ) of a function f at θ in the direction ϑ − θ to be provided this limit exists.

Sensitivity equations
In this section, we derive equations for the sensitivities of the solution (n e , n j , n a ) with respect to β and µ a .To simplify the notation, we use h to denote a given direction in the corresponding parameter space.We first want to derive the sensitivity of (n e , n j , n a ) with respect to β .Let (m e , m j , m a ) be the unique solution (guaranteed by results in [11]) of the initial boundary value problem Our aim here is to characterize the unique solution to (4.1) and to argue that m e = n eβ , m j = n j β and m a = n aβ , which implies that system (4.1) can be used to solve for the sensitivity of (n e , n j , n a ) with respect to β .