We study a general discrete planar system for modeling stage-structured populations. Our results include conditions for the global convergence of orbits to zero (extinction) when the parameters (vital rates) are time and density dependent. When the parameters are periodic we obtain weaker conditions for extinction. We also study a rational special case of the system for Beverton-Holt type interactions and show that the persistence equilibrium (in the positive quadrant) may be globally attracting even in the presence of interstage competition. However, we determine that with a sufficiently high level of competition, the persistence equilibrium becomes unstable (a saddle point) and the system exhibits period two oscillations.

Stage-structured models of single-species populations with lowest dimension in discrete time are expressed as planar systems of difference equations. For a general expression of these models, consider system

Under certain constraints on the various functions, including periodic vital rates and competition coefficients having the same common period

In this paper, we study the following abstraction of the matrix model (

System (

This is a system of type (

Another system of type (

The numbers

Also studied in [

Next, the model in [

We also mention the adult-juvenile model

We study the qualitative properties of the orbits of (

We also investigate convergence to zero with periodic parameters (extinction in a periodic environment). In particular, we show that convergence to zero occurs even if the mean value of

In the final section we study the dynamics of a rational special case of (

Discrete population models generally have been studied by numerous authors; see, for example, [

Conditions under which the orbits of (

Let

Let

Let

By (

By (

Next, applying Lemma

For functions

Theorem

However, if

In this section we obtain general sufficient conditions for the convergence of all orbits of the system to

We start with the following lemma; see [

Let

Note that (

Throughout this section we assume that

If the inequality

By (

Lemma

We consider an application of Theorem

Additionally, let

These inequalities ensure that the functions

Suppose that (

Note that (

Since in the above discussion the sequences

Corollary

In the autonomous case where the three parameter functions

If in Corollary

Assume that

Inequality (

These are real and a routine calculation shows that

Under suitable differentiability hypotheses, this inequality is implied by (

In the next and later sections it will be convenient to fold system (

The pair of first-order equations (

An even simpler reduction than the above is possible if

The results in this section show that global convergence to zero may occur even if (

The right-hand side of the above inequality is a linear expression. Consider the linear difference equation

By Lemma

Assume that

Suppose that the quadratic polynomial

System (

Each of (

Suppose that the numbers

if (

(a) Let

(b) Since

Now, the proof is completed by induction. The proof of (

Assume that (

(a) Equation (

(b) If

Hence,

(c) If

(a) Lemma

If

(b) To establish (

Since

Since

We claim that if

This claim is easily seen to be true by induction; we showed that it is true for

Given that

Upon rearranging terms and squaring,

(c) First, assume that

If

If

Some of the numbers

This is in fact true because

Let

In Lemma

Assume that (

Let

By induction it follows that

Note that (

In this section we apply some of the preceding results and obtain some new ones to study boundedness, extinction, and modes of survival in some rational special cases of (

For example, if we think of

We now examine boundedness and global convergence to 0 (extinction) in (

Assume that (

(a) Let the sequence

(b) Let the sequence

(a) By hypothesis, for all (large)

Next, let

By hypothesis, there is

Now an application of Corollary

(b) By (

The next result follows readily from Theorem

The origin

The above corollary is false when (

Consider system

Corollary

The following result is applicable to (

Assume that (

It is noteworthy that if in part (a) above

Corollary

Assume that (

We now explore the effects of competition in the autonomous system (

If

By the last two corollaries, all orbits of the rational system (

See (

With initial values

If

In addition, when

We note that

Let

Let

If we let

Now,

The rest of the proof follows from Lemma

Note that

If

Let

Since

If

The characteristic equation associated with the linearization of (

The roots of (

Since

This inequality holds, since

Note that when

Next,

We summarize the above results in the following lemma.

Let

locally asymptotically stable if and only if (

It is a saddle point if and only if (

Inequality (

Then

Since

(

From the above inequality we obtain

Thus if

Assume that (

Our final result establishes that when

The difference equation in (

Taking the difference of the right and left sides of (

When

Similarly, taking the sum of the right and left sides of (

Adding and subtracting

Simplifying the right hand side, it follows that

Now, since we are assuming that

From (

To ensure that

We summarize the above results as follows.

The second-order difference equation in (

The next result shows that a solution of period two appears when

The second-order difference equation in (

Suppose

Now

Adding and subtracting

Therefore,

Assume that (

Figure

Orbits illustrating period two oscillations and the saddle point.

We studied the dynamics of a general planar system that includes many common stage-structured population models that evolve in discrete time. Our hypotheses regarding system (

The authors declare that there is no conflict of interests regarding the publication of this paper.