Contemporary Mathematics
Existence-Uniqueness Results and Difference
Approximations for an Amphibian
Juvenile-Adult Model
Azmy S. Ackleh, Keng Deng and Qihua Huang
In honor of the 60th Birthday of Professor Simeon Reich
Abstract. We consider an amphibian population where individuals are
divided into two groups: juveniles (tadpoles) and adults (frogs). We assume
that juveniles are structured by age and adults are structured by size. Since
juveniles (tadpoles) live in water and adults (frogs) live on land, we assume
that competition occurs within stage only. This leads to a system of nonlinear
and nonlocal hyperbolic equations of first order. An explicit finite difference
approximation to this partial differential equation system is developed. Exis-
tence and uniqueness of the weak solution to the model are established and
convergence of the finite difference approximation to this unique solution is
1. Introduction
In this paper, we consider the dynamics of an amphibian population divided
into two groups 1) juveniles (tadpoles) and 2) adults (frogs). We assume that
juveniles are structured by their age while adults are structured by their size (since
often in such population adults become sexually mature when they reach a certain
length, e.g., see [18] for the green treefrogs). Let J(a, t) be the density of juveniles
of age a [0, amax] at time t [0, T ] and A(x, t) be the density of adults having size
x [xmin, xmax] at time t [0, T ]. Here, amax denotes the age at which a juvenile
(tadpole) metamorphoses into a frog (amax approximately equals five weeks for the
green treefrog [8, 12, 13, 16]), and xmin and xmax denote the minimum size and
the maximum size of a frog, respectively (green treefrog 15mm to 60mm [14]). Let
P (t) =
J(a, t)da be the total number of juveniles in the population at time t
and Q(t) =
A(x, t)dx be the total number of adults in the population at
time t. The function ν(a, t, P ) denotes the mortality rate of a juvenile of age a at
time t which depends on the number of tadpoles P due to competition for resources.
The function μ represents the mortality rate of an adult of size x, g represents the
2000 Mathematics Subject Classification. Primary 35L60, 65M06, 92D25.
This work was supported in part by NSF Grant #DMS-0718465.
c 2010 A.S. Ackleh, K. Deng and Q. Huang
Contemporary Mathematics
Volume 513, 2010
c 2010 A. S. Ackleh, K. Deng, and Q. Huang
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