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

proved.

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) =

amax

0

J(a, t)da be the total number of juveniles in the population at time t

and Q(t) =

xmax

xmin

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

1

Contemporary Mathematics

Volume 513, 2010

c 2010 A. S. Ackleh, K. Deng, and Q. Huang

1

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

proved.

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) =

amax

0

J(a, t)da be the total number of juveniles in the population at time t

and Q(t) =

xmax

xmin

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

1

Contemporary Mathematics

Volume 513, 2010

c 2010 A. S. Ackleh, K. Deng, and Q. Huang

1