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**Local existence for a general model of size-dependent population dynamics.**
*(English)*
Zbl 0931.92024

We are interested in a size structured population model with the growth rate depending on the individual’s size and time. There have been many investigations where the growth rate depends on the size. Recently, A. Calsina and J. Saldaña [J. Math. Biol. 33, No. 4, 335-364 (1995; Zbl 0828.92025)] have studied the case where the growth rate depends on the size as well as on the total population at each time. They have the model of plants in forests or plantations in their mind.

We are also motivated by the population model of the forest growth. In this case, the growth rate may be influenced by the environment such as light, temperature, and nutrients. These may change with time. It is also reasonable to think that the growth rate varies with the individual’s size of plants because the amount of light they capture may depend on it. From these points of view, it is natural to consider the growth rate depending on the size and time. We study the following initial-boundary value problem with nonlocal boundary conditions: \[ \begin{cases} u_t+ (V(x,t)u)_x= G(u(\cdot,t))(x), &x\in [0,l),\;a\leq t\leq T,\\ V(0,t)u(0,t)= C(t)+ F(u(\cdot,t)), &a\leq t\leq T,\\ u(x,a)= u_a(x), &x\in [0,l). \end{cases} \tag{SDP} \] Here \(a\geq 0\), \(l\in (0,\infty]\) is the maximum size, \(F\) and \(G\) are given mappings corresponding to birth and aging functions, respectively. The function \(V\) is the growth rate function depending on the size \(x\) and time \(t\) and the function \(C\) represents the inflow of zero-size individuals from an external source such as seeds carried by the wind or placed in a plantation. The unknown function \(u(x,t)\) stands for the density with respect to size \(x\) of a population at time \(t\). So the integral \(\int_{x_1}^{x_2} u(x,t) dx\) represents the number of individuals with size between \(x_1\) and \(x_2\) at time \(t\).

Our objective is to show the existence of a unique local solution and continuous dependence of the solution on the initial data.

We are also motivated by the population model of the forest growth. In this case, the growth rate may be influenced by the environment such as light, temperature, and nutrients. These may change with time. It is also reasonable to think that the growth rate varies with the individual’s size of plants because the amount of light they capture may depend on it. From these points of view, it is natural to consider the growth rate depending on the size and time. We study the following initial-boundary value problem with nonlocal boundary conditions: \[ \begin{cases} u_t+ (V(x,t)u)_x= G(u(\cdot,t))(x), &x\in [0,l),\;a\leq t\leq T,\\ V(0,t)u(0,t)= C(t)+ F(u(\cdot,t)), &a\leq t\leq T,\\ u(x,a)= u_a(x), &x\in [0,l). \end{cases} \tag{SDP} \] Here \(a\geq 0\), \(l\in (0,\infty]\) is the maximum size, \(F\) and \(G\) are given mappings corresponding to birth and aging functions, respectively. The function \(V\) is the growth rate function depending on the size \(x\) and time \(t\) and the function \(C\) represents the inflow of zero-size individuals from an external source such as seeds carried by the wind or placed in a plantation. The unknown function \(u(x,t)\) stands for the density with respect to size \(x\) of a population at time \(t\). So the integral \(\int_{x_1}^{x_2} u(x,t) dx\) represents the number of individuals with size between \(x_1\) and \(x_2\) at time \(t\).

Our objective is to show the existence of a unique local solution and continuous dependence of the solution on the initial data.

### MSC:

92D25 | Population dynamics (general) |

35A07 | Local existence and uniqueness theorems (PDE) (MSC2000) |

35Q92 | PDEs in connection with biology, chemistry and other natural sciences |