In this paper, we consider the following structured juvenile–adult population model:
equation(1.1)
View the MathML sourceJt(a,t)+Ja(a,t)+ν(a)J(a,t)=0,(a,t)∈(0,ā)×(0,T),At(x,t)+(g(x)A(x,t))x+μ(φ(t))A(x,t)=0,(x,t)∈(x¯,x̄)×(0,T),J(0,t)=∫x¯x̄β(φ(t))A(x,t)dx,t∈(0,T),g(x¯)A(x¯,t)=J(ā,t),t∈(0,T),J(a,0)=J0(a),a∈[0,ā],A(x,0)=A0(x),x∈[x¯,x̄],
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where J(a,t)J(a,t) and A(x,t)A(x,t) denote the density of juveniles of age aa and adults of size xx at time tt, respectively, View the MathML sourceā denotes the age at which juveniles metamorphose into adults of minimum size View the MathML sourcex¯, and View the MathML sourcex̄ denotes the maximum size of adults. The function View the MathML sourceφ(t)=∫x¯x̄A(x,t)dx is the total population of adults. The parameters νν and μμ are the mortality rates for juveniles and adults, respectively. The functions gg and ββ are the growth and reproduction rates for adults, respectively. Motivated by an amphibian population of green tree frogs (Hyla cinerea), we recently developed such a model in [1]. We assumed that juveniles live in an environment with abundant resources and thus do not compete, while adults live in an environment with limiting resource and thus competition between them takes place. We then established existence–uniqueness results and discussed the long-time behavior of the solution of the model via a comparison principle.
Our objective here is to conduct sensitivity analysis for model (1.1). The importance of sensitivity equations has long been recognized as they provide a measure of model response (output) to variation in the underlying model parameters (e.g., see [2] and [3] and the references therein). The derivation of sensitivity equations for discrete structured population models (matrix models) has received a great deal of attention in the past few decades (see [2] and the many references therein). However, little work has been done on the derivation of sensitivity equations for continuous structured population models which are of Mckendrick–von Foerster partial differential equations type. Such equations are useful for computing variances of estimated model parameters from observation data (e.g., see, [4], [5], [6] and [7]). Our motivation comes from paper [8], wherein sensitivity partial differential equations were derived for the following linear size-structured population model:
equation(1.2)
View the MathML sourceut(x,t)+(g(x)u(x,t))x+μ(x)u(x,t)=0,(x,t)∈(0,x̄)×(0,T),g(0)u(0,t)=∫0x̄β(x)u(x,t)dx,t∈(0,T),u(x,0)=u0(x),x∈[0,x̄].
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There are two main differences between models (1.2) and (1.1). First, (1.2) is a single equation model, but (1.1) is a system of coupled equations. Second, the vital rates in (1.2) are linear, but the reproduction and mortality rates for adults in (1.1) are dependent on the total population of adults. Due to the different structure of model (1.1), the situation becomes more complicated, and certain techniques used for (1.2) seem not applicable to (1.1).
The paper is organized as follows. In Section 2, we establish an existence result for directional derivatives with respect to parameters. In Section 3, we derive sensitivity partial differential equations for the sensitivities of the solution with respect to the reproduction and mortality rates for adults. In Section 4, we make numerical simulations to apply these equations to the population of green tree frogs.
