Existence and uniqueness of solutions for a hierarchical system of two age-structured populations

Abstract

We propose a class of new hierarchical model for the evolution of two interacting age-structured populations, which is a system of integro-partial differential equations with global feedback boundary conditions and may describe the interactions such as competition, cooperation and predator-prey relation. Based upon a group of natural conditions, the existence and uniqueness of solutions on infinite time interval are proved by means of fixed point and extension principle, and the continuous dependence of the solution on the initial age distribution is established. These results lay a sound basis for the investigation of stability, controllability and variable optimal control problems.

Keywords

Hierarchy of ages / population system / integro-partial differential equations / existence and uniqueness / fixed points

1 Introduction

A history of ecological investigations in past 80 years shows that there exist differences of dominance ranks (social status) among individuals in most of biological populations, which have significant effects on individual's vital parameters and population evolutions, see [8] and more than 200 references cited therein. On the other hand, quantitative studies for hierarchical populations by mathematical modelling appear recently, see [1, 2, 4–7, 9–12, 14–18]. All of these works focus on single populations, researches on interacting multi-population systems are quite rare. Although the results for the evolution of single population may be helpful for the exploration of multi-population systems, actual habitats are often shared by many populations. Therefore investigations of hierarchical multi-population systems are both of practical significance and more challenging as well.

It is well known that a careful study of systems of two populations is essential in the process of analyzing multi-population systems. Based upon the dominance rank of individuals, we in this article propose a general model for interacting two populations, which may describe the interaction of competition, cooperation and predation between the two populations, and show the complicated relations among the individuals inside each population. The purpose of the present paper is, under a group of weak and natural conditions on the parameters, to establish the existence, uniqueness, non-negativity, boundedness, and continuous dependence of solutions on initial distributions to the system model of integro-partial differential equations. The results obtained will pave the way to stability, controllability and optimal control problems of the hierarchical population system.

2 System model

Consider two biological populations surviving in an ecological environment, the interaction between them may be cooperation, competition, predation, or some complicated types; and similar complex relations among the individuals within each population may also be observed. Suppose that the difference of dominance rank is formed by age according to the principle: the vital rates (e.g., fertility, mortality) of an individual of age a depend mainly on the size of individuals with age larger than a, and less on others. We formulate the following system model (i=1,2):

\begin{array}{l}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }a}}=-[{\mu }_1(a)+m_1(E(p_1)(a,t))+f_1(E(p_2)(a,t))]p_1(a,t),(a,t)\in Q_{1T},\\ {\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }a}}=-[{\mu }_2(a)+m_2(E(p_2)(a,t))+f_2(E(p_1)(a,t))]p_2(a,t),(a,t)\in Q_{2T},\\ p_i(0,t)={\displaystyle {\int }_0^{A_i}{\beta }_i(a,E(p_i)(a,t))p_i(a,t)\mathrm{d}a,t\in (0,T),}\\ p_i(a,0)=p_i^0(a),a\in [0,A_i],\\ E(p_i)(a,t)=\alpha {\displaystyle {\int }_0^a{p}_i(r,t)\mathrm{d}r+{\displaystyle {\int }_a^{A_i}{p}_i(r,t)\mathrm{d}r,(a,t)\in Q_{iT},}}\end{array}\end{array}

where Q_{iT}=(0,A_i)\times (0,T), A_i is the maximal age of individuals in the population i, and T the horizon. Other functions and parameters mean as follows (i,j=1,2,j\ne i):

p_i(a,t): the density of population i of age a at time t;

{\mu }_i(a): natural death rate of population i;

m_i(E(p_i)(a,t)): interaction of individuals in the population i;

f_i(E(p_j)(a,t)): effect of population j on population i;

{\beta }_i(a,E(p_i)(a,t)): the average fertility of population i;

p_i^0(a): the initial distribution of population i;

E(p_i)(a,t): the internal environment in population i facing individuals of

age a at the moment t; denotes the hierarchy efficient.

For the sake of theoretic analysis, we normalize the above model. Let A=max(A_1,A_2), Q_T=(0,A)\times (0,T), and zero extensions should be made if necessary. The above model can be rewritten as

(1)\begin{array}{l}\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }a}}=-[{\mu }_1(a)+m_1(E(p_1)(a,t))+f_1(E(p_2)(a,t))]p_1(a,t),(a,t)\in Q_T,\\ {\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }a}}=-[{\mu }_2(a)+m_2(E(p_2)(a,t))+f_2(E(p_1)(a,t))]p_2(a,t),(a,t)\in Q_T,\\ p_i(0,t)={\displaystyle {\int }_0^A{\beta }_i(a,E(p_i)(a,t))p_i(a,t)\mathrm{d}a,t\in (0,T),}\\ p_i(a,0)=p_i^0(a),a\in [0,A],\\ E(p_i)(a,t)=\alpha {\displaystyle {\int }_0^a{p}_i(r,t)\mathrm{d}r+{\displaystyle {\int }_a^A{p}_i(r,t)\mathrm{d}r,(a,t)\in Q_T.}}\end{array}\end{array}

In this paper, we make the following assumptions (i=1,2):

(A1) 0⩽{\beta }_i(a,s)⩽{\beta }_i^{\ast } for all (a,s)\in [0,A]\times [0,+\mathrm{\infty }), and {\beta }_i^{\ast } is constant. For any given M>0, there is L(M)>0 such that

|{\beta }_i(a,s_1)-{\beta }_i(a,s_2)|⩽L(M)|s_1-s_2|\mathrm{i}\mathrm{f}{\mathrm{s}}_1,{\mathrm{s}}_2\in [0,\mathrm{M}];

(A2) {\mu }_i(a)>0,{\int }_0^A{\mu }_i(a)\mathrm{d}a=+\mathrm{\infty };

(A3) \mid m_i(s)\mid ⩽m_i^{\ast } for all s\in [0,+\mathrm{\infty }), and m_i^{\ast } is constant; m_i is locally Lipschitz continuous;

(A4) \mid f_i(s)\mid ⩽f_i^{\ast } for all s\in [0,+\mathrm{\infty }), and f_i^{\ast } is constant; f_i is locally Lipschitz continuous;

(A5) 0⩽p_i^0(a)⩽p_i^{\ast } for all a\in [0,A], and p_i^{\ast } is constant.

3 Existence and uniqueness of solutions

Definition 3.1 p(a,t)=(p_1(a,t),p_2(a,t))\in [L^1((0,A)\times (0,T)){]}^2 is call to be a solution to (1) if it is absolutely continuous at every characteristic line a-t=c (with c constant) and

\{\begin{array}{l}D{p}_1(a,t)+[{\mu }_1(a)+m_1(E(p_1)(a,t))+f_1(E(p_2)(a,t))]p_1(a,t)=0,\mathrm{a}.\mathrm{e}.(a,t)\in Q_T,\\ D{p}_2(a,t)+[{\mu }_2(a)+m_2(E(p_2)(a,t))+f_2(E(p_1)(a,t))]p_2(a,t)=0,\mathrm{a}.\mathrm{e}.(a,t)\in Q_T,\\ \underset{\epsilon \to 0}{lim}{p}_i(\epsilon ,t+\epsilon )={\displaystyle {\int }_0^A{\beta }_i(a,E(p_i)(a,t))p_i(a,t)\mathrm{d}a,\mathrm{a}.\mathrm{e}.t\in (0,T),i=1,2,}\\ \underset{\epsilon \to 0}{lim}{p}_i(a+\epsilon ,\epsilon )=p_i^0(a),\mathrm{a}.\mathrm{e}.a\in [0,A],i=1,2,\\ E(p_i)(a,t)={\alpha }_i{\displaystyle {\int }_0^a{p}_i(r,t)\mathrm{d}r+{\displaystyle {\int }_a^A{p}_i(r,t)\mathrm{d}r,\mathrm{a}.\mathrm{e}.(a,t)\in Q_T,i=1,2,}}\end{array}

where

D{p}_i(a,t)=\underset{\epsilon \to 0}{lim}{\epsilon }^{-1}[p_i(a+\epsilon ,t+\epsilon )-p_i(a,t)],i=1,2.

In what follows, we establish the existence and uniqueness result for solutions to (1) by means of the principle of contracting mappings.

Firstly, let p_i in the functions m_i,f_i,{\beta }_i be fixed with a non-negative function q_i,i=1,2. Then we have the following linear model:

(2)\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_1}{\mathrm{\partial }a}}=-[{\mu }_1(a)+m_1(E(q_1)(a,t))+f_1(E(q_2)(a,t))]p_1(a,t),(a,t)\in Q_T,\\ {\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }t}}+{\displaystyle \frac{\mathrm{\partial }{p}_2}{\mathrm{\partial }a}}=-[{\mu }_2(a)+m_2(E(q_2)(a,t))+f_2(E(q_1)(a,t))]p_2(a,t),(a,t)\in Q_T,\\ p_i(0,t)={\displaystyle {\int }_0^A{\beta }_i(a,E(q_i)(a,t))p_i(a,t)\mathrm{d}a,t\in (0,T),}\\ p_i(a,0)=p_i^0(a),a\in [0,A],\\ E(q_i)(a,t)=\alpha {\displaystyle {\int }_0^a{q}_i(r,t)\mathrm{d}r+{\displaystyle {\int }_a^A{q}_i(r,t)\mathrm{d}r,(a,t)\in Q_T.}}\end{array}

According to results in [13], there is a unique solution p(a,t;q),q=(q_1,q_2), to system (2), which is non-negative and bounded. By the method of characteristic lines, one sees

(3)

where

(4)\begin{array}{rl}& {\mathrm{\Pi }}_i(a,t,s;q)\\ & =\exp \{-{\int }_0^s[{\mu }_i(a-\tau )+m_i(E(q_i)(a-\tau ,t-\tau ))\\ & +f_i(E(q_j)(a-\tau ,t-\tau ))]\mathrm{d}\tau \},i,j=1,2;i\ne j,s\in (0,min\{a,t\}).\end{array}

Using (3) and the third equation in system (2), we claim that b_i(t;q):=p_i(0,t;q) (the fertility of population i) solves the following Volterra integral equation

(5)b_i(t;q)=F_i(t;q)+{\int }_0^t{K}_i(t,s;q)b_i(t-s;q)\mathrm{d}s,t\in (0,T),i=1,2,

where F_i and K_i are given by

(6)F_i(t;q)=\{\begin{array}{l}{\displaystyle {\int }_t^A{\beta }_i(a,E(q_i)(a+t,t))p_i^0(a){\mathrm{\Pi }}_i(a+t,t,t;q)\mathrm{d}a,\mathrm{a}.\mathrm{e}.t\in (0,c);}\\ 0,t⩾c,\end{array}

(7)\begin{array}{l}{K}_i(t,a;q)=\{\begin{array}{l}{\beta }_i(a,E(q_i)(a,t)){\mathrm{\Pi }}_i(a,t,a;q),\mathrm{a}.\mathrm{e}.(a,t)\in Q_T;\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}\end{array}

in which c=min\{A,T\}.

In what follows, we carefully deal with the case T>A. The other case can be treated similarly.

Assumptions (A1)–(A5) lead to

(8)0⩽K_i(t,a;q)⩽{\beta }_i^{\ast }\exp \{A(f_i^{\ast }+m_i^{\ast })\}=:K_i^{\ast }\mathrm{a}.\mathrm{e}.(a,t)\in Q_T,

(9)\begin{array}{l}\begin{array}{l}0⩽F_i(t;q)⩽A{\beta }_i^{\ast }{p}_i^{\ast }\exp \{A(f_i^{\ast }+m_i^{\ast })\}=:F_i^{\ast }\mathrm{a}.\mathrm{e}.t\in (0,T).\end{array}\end{array}

Let M_T=T{F}_i^{\ast }\exp \{A(f_i^{\ast }+m_i^{\ast })+T{K}_i^{\ast }\}+A{p}_i^{\ast }\exp \{A(f_i^{\ast }+m_i^{\ast })\}, and define the set

H=\{(v_1,v_2)\in [L^{\mathrm{\infty }}(0,T;L^1(0,A)){]}^2;v_i(a,t)⩾0\mathrm{a}.\mathrm{e}.(a,t)\in Q_T,

{\Vert v_i(\cdot ,t)\Vert }_{L^1(0,A)}⩽M_T\mathrm{a}.\mathrm{e}.t\in (0,T),i=1,2\},

where \Vert (v_1,v_2)\Vert =\Vert v_1\Vert +\Vert v_2\Vert.

Lemma 3.1　 There are constants M_{iT}>0, such that, for all q^i=(q_1^i,q_2^i)\in H,i=1,2, a.e. t\in (0,T), the following inequalities hold:

(10)\begin{array}{rl}& |F_i(t;q^1)-F_i(t;q^2)|\\ & ⩽M_{1T}\left({\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}+{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\right),\end{array}

(11)\begin{array}{rl}& |b_i(t;q^1)-b_i(t;q^2)|\\ & ⩽M_{2T}\left({\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}+{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\right),\end{array}

(12)\begin{array}{l}0⩽b_i(t;q^j)⩽M_{2T},j=1,2.\end{array}

Proof　For all q^i\in H,i=1,2, it is followed by (6) that F_i(t;q^j)\equiv 0 if t\in (A,T), and relation (10) is true.

If t\in (0,A), then by (6) one has

(13)\begin{array}{rl}& \left|F_1(t;q^1)-F_1(t;q^2)\right|\\ & ⩽{\int }_0^{\mathrm{\infty }}\left|{\beta }_1(a+t,E(q_1^1)(a+t,t))-{\beta }_1(a+t,E(q_1^2)(a+t,t))\right|\cdot p_1^0(a){\mathrm{\Pi }}_1(a+t,t,t;q^1)\mathrm{d}a\\ & +{\int }_0^{\mathrm{\infty }}{\beta }_1(a+t,E(q_1^2)(a+t,t))p_1^0(a)\cdot \left|{\mathrm{\Pi }}_1(a+t,t,t;q^1)-{\mathrm{\Pi }}_1(a+t,t,t;q^2)\right|\mathrm{d}a.\end{array}

From assumptions and (13), it follows that, for t\in (0,T),

(14)\begin{array}{rl}& \left|F_1(t;q^1)-F_1(t;q^2)\right|\\ & ⩽A{p}_1^{\ast }L(M_T){\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}\\ & +A{p}_1^{\ast }{\beta }_1^{\ast }L(M_T){\int }_0^t\left|E(q^1)(a+s,s)-E(q^2)(a+s,s)\right|\mathrm{d}s.\end{array}

The definition of E(p_i) implies that

(15)\begin{array}{rl}& {\int }_0^t\left|E(q_1^1)(a+s,s)-E(q_1^2)(a+s,s)\right|\mathrm{d}s\\ & ⩽{\int }_0^t\left\{\alpha {\int }_0^{a+s}\left|q_1^1(r,s)-q_1^2(r,s)\right|\mathrm{d}r+{\int }_{a+s}^A\left|q_1^1(r,s)-q_1^2(r,s)\right|\mathrm{d}r\right\}\mathrm{d}s\\ & ⩽{\int }_0^t{\int }_0^A\left|q_1^1(r,s)-q_1^2(r,s)\right|\mathrm{d}r\mathrm{d}s\\ & ⩽{\int }_0^t{\Vert q_1^1(\cdot ,s)-q_1^2(\cdot ,s)\Vert }_{L^1(0,A)}\mathrm{d}s.\end{array}

Substituting (15) into (14), one gets

\begin{array}{rl}& \left|F_1(t;q^1)-F_1(t;q^2)\right|\\ & ⩽A{p}_1^{\ast }L(M_T)max\left\{1,{\beta }_1^{\ast }\right\}\\ & \cdot \left({\Vert q_1^1(\cdot ,t)-q_1^2(\cdot ,t)\Vert }_{L^1(0,A)}+{\int }_0^t{\Vert q_1^1(\cdot ,s)-q_1^2(\cdot ,s)\Vert }_{L^1(0,A)}\mathrm{d}s\right).\end{array}

Thus, inequality (10) is true for i=1 and M_{1T}=A{p}_1^{\ast }L(M_T)max\left\{1,{\beta }_1^{\ast }\right\}. The proof for i=2 is similar.

The existence and uniqueness of non-negative and continuous solutions to equation (5) can be shown in a similar way as in [3]. By (5), (8)−(9) and Bellmann’s inequality, it is easy to obtain b_1(t;q)⩽b_1^{\ast } (with b_1^{\ast } positive constant). Next we verify inequality (11).

From (5) it is deduced that

(16)\begin{array}{rl}& \left|b_1(t;q^1)-b_1(t;q^2)\right|\\ & ⩽\left|F_1(t;q^1)-F_1(t;q^2)\right|+{\int }_0^t\left|K_1(t,t-s;q^1)-K_1(t,t-s;q^2)\right|b_1(s;q^1)\mathrm{d}s\\ & +{\int }_0^t{K}_1(t,t-s;q^2)\left|b_1(s;q^1)-b_1(s;q^2)\right|\mathrm{d}s\\ & ⩽M_{1T}({\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}+{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s)\\ & +b_1^{\ast }L(M_T)T({\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}\\ & +{\beta }_1^{\ast }{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s)\\ & +K_1^{\ast }{\int }_0^t\left|b_1(s;q^1)-b_1(s;q^2)\right|\mathrm{d}s\\ & ⩽M_{3T}D(t)+K_1^{\ast }{\int }_0^t\left|b_1(s;q^1)-b_1(s;q^2)\right|\mathrm{d}s,\end{array}

where M_{3T}=M_{1T}+b_1^{\ast }L(M_T)Tmax\{1,{\beta }_1^{\ast }\}, and

D(t)={\Vert q^1(\cdot ,t)-q^2(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}+{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s.

Combining (16) with Gronwall’s inequality, one sees

\begin{array}{rl}& |b_1(t;q^1)-b_1(t;q^2)|⩽M_{3T}D(t)+M_{3T}{\int }_0^t{K}_1^{\ast }D(s)\exp \left\{{\int }_s^t{K}_1^{\ast }\mathrm{d}\tau \right\}\mathrm{d}s\\ & ⩽M_{3T}D(t)+M_{3T}{\int }_0^t{K}_1^{\ast }D(s)\exp \{K_1^{\ast }T\}\mathrm{d}s\\ & =M_{3T}D(t)+M_{3T}{K}_1^{\ast }\exp (K_1^{\ast }T){\int }_0^t\{{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\\ & +{\int }_0^s{\Vert q^1(\cdot ,\tau )-q^2(\cdot ,\tau )\Vert }_{[L^1(0,A){]}^2}\mathrm{d}\tau \}\mathrm{d}s\\ & ⩽M_{3T}D(t)+M_{3T}{K}_1^{\ast }\exp (K_1^{\ast }T)(1+T){\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\\ & ⩽M_{3T}(1+K_1^{\ast }\exp (K_1^{\ast }T)(1+T))D(t),\end{array}

which is the conclusion for i=1 and M_{2T}=M_{3T}max\left\{1,K_1^{\ast }\exp (K_1^{\ast }T)(1+T)\right\}. The proof for the case i=2 is similar.

Secondly, define the mapping G as follows: for q=(q_1,q_2)\in H, let

(Gq)(a,t)=p(a,t;q)=(p_1(a,t;q),p_2(a,t;q)),

where (p_1(a,t;q),p_2(a,t;q)) (given by (3)) solves model (2).

By relations (3)−(9) it is not difficult to show G(q)\in H for each q\in H. The following lemma is a key technical result.

Lemma 3.2　 There is a constant M_{4T}>0, such that, for all q^i=(q_1^i,q_2^i)\in H,i=1,2, a.e. t\in (0,T), the following holds:

(17){\Vert (G{q}^1)(\cdot ,t)-(G{q}^2)(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}⩽M_{4T}{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s.

Proof　From the definition of the mapping G and relations (3)–(5), it follows that, for t\in (0,A),

(18)\begin{array}{rl}& {\Vert (G{q}^1)(\cdot ,t)-(G{q}^2)(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}\\ & ={\Vert \left(\begin{array}{c}{p}_1^1(\cdot ,t;q^1)\\ p_2^1(\cdot ,t;q^1)\end{array}\right)-\left(\begin{array}{c}{p}_1^2(\cdot ,t;q^2)\\ p_2^2(\cdot ,t;q^2)\end{array}\right)\Vert }_{[L^1(0,A){]}^2}\\ & ={\Vert p_1^1(\cdot ,t;q^1)-p_1^2(\cdot ,t;q^2)\Vert }_{L^1(0,A)}+{\Vert p_2^1(\cdot ,t;q^1)-p_2^2(\cdot ,t;q^2)\Vert }_{L^1(0,A)}\\ & ⩽{\int }_0^t\left|b_1(t-a;q^1)-b_1(t-a;q^2)\right|{\mathrm{\Pi }}_1(a,t,a;q^1)\mathrm{d}a\\ & +{\int }_0^t{b}_1(t-a;q^2)\left|{\mathrm{\Pi }}_1(a,t,a;q^1)-{\mathrm{\Pi }}_1(a,t,a;q^2)\right|\mathrm{d}a\\ & +{\int }_t^A{p}_1^0(a-t)\left|{\mathrm{\Pi }}_1(a,t,t;q^1)-{\mathrm{\Pi }}_1(a,t,t;q^2)\right|\mathrm{d}a\\ & +{\int }_0^t\left|b_2(t-a;q^1)-b_2(t-a;q^2)\right|{\mathrm{\Pi }}_2(a,t,a;q^1)\mathrm{d}a\\ & +{\int }_0^t{b}_2(t-a;q^2)\left|{\mathrm{\Pi }}_2(a,t,a;q^1)-{\mathrm{\Pi }}_2(a,t,a;q^2)\right|\mathrm{d}a\\ & +{\int }_t^A{p}_2^0(a-t)\left|{\mathrm{\Pi }}_2(a,t,t;q^1)-{\mathrm{\Pi }}_2(a,t,t;q^2)\right|\mathrm{d}a.\end{array}

Combining (18) with Lemma 3.1 and assumptions (A1)–(A5), we arrive at

\begin{array}{rl}& {\Vert (G{q}^1)(\cdot ,t)-(G{q}^2)(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}\\ & ⩽2M_{2T}{\int }_0^t\{{\Vert q^1(\cdot ,t-a)-q^2(\cdot ,t-a)\Vert }_{[L^1(0,A){]}^2}\\ & +{\int }_0^{t-a}{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\}\mathrm{d}a\\ & +2M_{2T}L(M_T){\int }_0^t{\int }_0^a\left|E(q^1)(a-\tau ,t-\tau )-E(q^2)(a-\tau ,t-\tau )\right|\mathrm{d}\tau \mathrm{d}a\\ & +L(M_T){\int }_t^A{p}_1^0(a-t){\int }_0^t\left|E(q^1)(a-\tau ,t-\tau )-E(q^2)(a-\tau ,t-\tau )\right|\mathrm{d}\tau \mathrm{d}a\\ & +L(M_T){\int }_t^A{p}_2^0(a-t){\int }_0^t\left|E(q^1)(a-\tau ,t-\tau )-E(q^2)(a-\tau ,t-\tau )\right|\mathrm{d}\tau \mathrm{d}a\\ & ⩽2M_{2T}(1+T){\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\\ & +2M_{2T}L(M_T)T{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\\ & +L(M_T)A(p_1^{\ast }+p_2^{\ast }){\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\\ & ⩽M_{4T}{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s,\end{array}

where M_{4T}=2M_{2T}[1+T+TL(M_T)]+L(M_T)A(p_1^{\ast }+p_2^{\ast }). Hence, the conclusion is correct for t\in (0,A). The proof for the case t\in (A,T) is similar

We are now ready to present one of the main results in this paper.

Theorem 3.1　 If (A1)–(A5) hold, then model system (1) admits a unique solution p(a,t),(a,t)\in Q_T, which is non-negative and bounded.

Proof　Let \lambda >M_{4T}. Define the equivalent norm in space [L^{\mathrm{\infty }}(0,T;L^1(0,A)){]}^2 as follows:

\Vert q{\Vert }_{\ast }=Esssu{p}_{t\in (0,T)}\left\{e^{-\lambda t}\Vert q{\Vert }_{[L^1(0,A){]}^2}\right\}.

From Lemma 3.2, one derives

\begin{array}{rl}& \Vert G{q}^1-G{q}^2{\Vert }_{\ast }\\ & =Esssu{p}_{t\in (0,T)}\left\{e^{-\lambda t}{\Vert (G{q}^1)(\cdot ,t)-(G{q}^2)(\cdot ,t)\Vert }_{[L^1(0,A){]}^2}\right\}\\ & ⩽M_{4T}Esssu{p}_{t\in (0,T)}\left\{e^{-\lambda t}{\int }_0^t{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\right\}\\ & ⩽M_{4T}Esssu{p}_{t\in (0,T)}\left\{e^{-\lambda t}{\int }_0^t{e}^{\lambda s}{e}^{-\lambda s}{\Vert q^1(\cdot ,s)-q^2(\cdot ,s)\Vert }_{[L^1(0,A){]}^2}\mathrm{d}s\right\}\\ & ⩽\frac{M_{4T}}{\lambda }\Vert q^1-q^2{\Vert }_{\ast }.\end{array}

Therefore, G is a contraction on ([L^{\mathrm{\infty }}(0,T;L^1(0,A)){]}^2,\Vert \cdot {\Vert }_{\ast }). The Banach principle assures that G owns a unique fixed point, which is the solution to system (1).

On the other hand, the solution must be non-negative and bounded by (3) and (12). The proof is completed.

Note that T is an arbitrary constant and the boundedness of solutions, the principle of continuations implies the following global result.

Corollary 3.1　 There exists a unique non-negative solution to (1) on (0,+\mathrm{\infty }).

4 Continuous dependence of solutions on initial distributions

Theorem 4.1　 Solutions of model (1) are uniformly continuous with respect to the initial data p^0=(p_1^0,p_2^0).

Proof　Since the solution to (1) is just the fixed point of G, relations (3)−(7) can be rewritten as follows:

(19)

where

(20)\begin{array}{rl}& {\mathrm{\Pi }}_i(a,t,s)\\ & =\exp \{-{\int }_0^s[{\mu }_i(a-\tau )+m_i(E(p_i)(a-\tau ,t-\tau ))\\ & +f_i(E(p_j)(a-\tau ,t-\tau ))]\mathrm{d}\tau \},i,j=1,2;i\ne j,s\in (0,min\{a,t\});\end{array}

(21)b_i(t)=F_i(t)+{\int }_0^t{K}_i(t,s)b_i(t-s)\mathrm{d}s,t\in (0,T),i=1,2,

in which F_i and K_i are given by

(22)\begin{array}{l}\begin{array}{l}{F}_i(t)=\{\begin{array}{l}{\displaystyle {\int }_t^A{\beta }_i(a,E(p_i)(a+t,t))p_i^0(a){\mathrm{\Pi }}_i(a+t,t,t)\mathrm{d}a,\mathrm{a}.\mathrm{e}.t\in (0,c);}\\ 0,t⩾c,\end{array}\end{array}\end{array}

(23)\begin{array}{l}{K}_i(t,a)=\{\begin{array}{l}{\beta }_i(a,E(p_i)(a,t)){\mathrm{\Pi }}_i(a,t,a),\mathrm{a}.\mathrm{e}.(a,t)\in Q_T;\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e},\end{array}\end{array}

where c=min\{A,T\}.

Let p^i(a,t)=(p_1^i(a,t),p_2^i(a,t)) be the solution of (1) corresponding to the initial distribution p^{0i}(a)=(p_1^{0i}(a),p_2^{0i}(a)), i=1,2.

If a⩾t, then it follows from (19)−(20) that

(24)\begin{array}{rl}& \mid p_1^1(a,t)-p_1^2(a,t)\mid =\mid p_1^{01}(a-t){\mathrm{\Pi }}_1^1(a,t,t)-p_1^{02}(a-t){\mathrm{\Pi }}_1^2(a,t,t)\mid \\ & =|p_1^{01}(a-t)\exp \{-{\int }_0^t[{\mu }_1(a-\tau )+m_1(E(p_1^1)(a-\tau ,t-\tau ))\\ & +f_1(E(p_2^1)(a-\tau ,t-\tau ))]\mathrm{d}\tau \}\\ & -p_1^{02}(a-t)\exp \{-{\int }_0^t[{\mu }_1(a-\tau )+m_1(E(p_1^2)(a-\tau ,t-\tau ))\\ & +f_1(E(p_2^2)(a-\tau ,t-\tau ))]\mathrm{d}\tau \}|\\ & ⩽|p_1^{01}(a-t)\exp \left\{-{\int }_0^t[m_1(E(p_1^1)(a-\tau ,t-\tau ))+f_1(E(p_2^1)(a-\tau ,t-\tau ))]\mathrm{d}\tau \right\}\\ & -p_1^{02}(a-t)\exp \left\{-{\int }_0^t[m_1(E(p_1^2)(a-\tau ,t-\tau ))+f_1(E(p_2^2)(a-\tau ,t-\tau ))]\mathrm{d}\tau \right\}|\\ & ⩽|p_1^{01}(a-t)-p_1^{02}(a-t)|\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}\\ & +p_1^{02}(a-t)|\exp \left\{-{\int }_0^t[m_1(E(p_1^1)(a-\tau ,t-\tau ))+f_1(E(p_2^1)(a-\tau ,t-\tau ))]\mathrm{d}\tau \right\}\\ & -\exp \left\{-{\int }_0^t[m_1(E(p_1^2)(a-\tau ,t-\tau ))+f_1(E(p_2^2)(a-\tau ,t-\tau ))]\mathrm{d}\tau \right\}|\\ & ⩽\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}|p_1^{01}(a-t)-p_1^{02}(a-t)|\\ & +p_1^{\ast }\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}{\int }_0^t[|m_1(E(p_1^1)(a-\tau ,t-\tau ))-m_1(E(p_1^2)(a-\tau ,t-\tau ))|\\ & +|f_1(E(p_2^1)(a-\tau ,t-\tau ))-f_1(E(p_2^2)(a-\tau ,t-\tau ))|]\mathrm{d}\tau \\ & ⩽\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}|p_1^{01}(a-t)-p_1^{02}(a-t)|\\ & +2p_1^{\ast }L(M_T)\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}{\int }_0^t[|E(p_1^1)(a-\tau ,t-\tau )-E(p_1^2)(a-\tau ,t-\tau )|\\ & +|E(p_2^1)(a-\tau ,t-\tau )-E(p_2^2)(a-\tau ,t-\tau )|]\mathrm{d}\tau \\ & ⩽\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}|p_1^{01}(a-t)-p_1^{02}(a-t)|\\ & +2p_1^{\ast }L(M_T)\exp \left\{T(m_1^{\ast }+f_i^{\ast })\right\}{\int }_0^t[\alpha {\int }_0^{a-\tau }|p_1^1(r,t-\tau )-p_1^2(r,t-\tau )|\mathrm{d}r\\ & +{\int }_{a-\tau }^A|p_2^1(r,t-\tau )-p_2^2(r,t-\tau )|\mathrm{d}r]\mathrm{d}\tau \\ & ⩽C\Vert p^{01}-p^{02}{\Vert }_{L^{\mathrm{\infty }}[0,A]},\end{array}

where C is a positive constant.

If , then inequality (24) is also true with another constant C. In a word, there must be a constant C such that

\Vert p^1-p^2{\Vert }_{L^{\mathrm{\infty }}(Q_T)}⩽C\Vert p^{01}-p^{02}{\Vert }_{L^{\mathrm{\infty }}[0,A]},

which implies the uniform continuity of solutions to (1) with respect to the initial distributions. The proof is completed.

5 Concluding remarks

For the model system of integro-partial differential equations in Section 2, we have proved the existence and uniqueness of non-negative solutions by means of the principle of contraction mappings. A problem followed closely by is: how to find the solution if one knows the model parameters? It is not difficult to understand that finding an exact analytical formula for the solution is almost impossible, because model (1) is a strongly coupled nonlinear system. Although the principle of contraction mappings provides us an approximation method, the convergence rate is often slow. Other fast converging numerical algorithms are needed, which is one of our ongoing works.

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