On global stability of discrete population models

On Global Stability of Discrete Populatlon Models

GERD ROSENKRANZ Unioerkit Heidelberg, Sonderforschungsbereich 123, Im Neuenheimer Feld 293, D -6900 Heidelberg 1, West Germany Received 30 September 1982; revised 28 Februaty I983

ABSTRACT A necessary and sufficient condition for the global stability of a large class of discrete population models is provided which does not require the construction of a Liapunov function. The general result is applied to difference equations defined in terms of “two hump”

functions

and to an example

of frequency

dependent

selection.

INTRODUCTION First order difference equations provide a general framework for mathematical models of ecological systems which develop at discrete time intervals rather than as a continuous process. The dynamical behavior of such models, leading from stability to “chaos” if some parameters are varied in an appropriate way, has been studied in articles dealing with special models in biology (May [4-61, Nobile et al. [7]) as well as in monographs on recurrences and iterated maps (Gumowski and Mira [3], Collet and Eckmamr [l]). In contrast with the abovementioned work, the present paper establishes conditions under which the equilibrium point of a one-dimensional population model &I+1

=f(xn)

(1)

is globally stable. Here x, represents a characteristic feature of the population in the n th generation such as population size, frequency of a certain gene, and so on. An equilibrium point of (1) is a point x* with

x* =f(x*), i.e. a fixpoint off. some neighborhood MATHEMATICAL

We say that x* is locally stable if x, + x* for all x0 in of x*, and globally stable if convergence to x* takes place BIOSCIENCES

641221-23 1 (1983)

DElsevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017

227 0025-5564/83/$03.00

GERD

228

ROSENKRANZ

for all 0 < x0 < 1. (We tacitly assume that f is a nonnegative real valued function on [0,11.) Usually, global stability of an equilibrium point for a difference equation is proved via the construction of a Liapunov function. This general method has the disadvantage that it is not obvious how to construct such a function for a particular model. Therefore we try to use the properties off directly to determine global stability. Cull [2] presents a necessary and sufficient criterion to check global stability of population models having the following property (among others): If f has a maximum in 0 < X~ < x*, then f is monotonically decreasing for x > x~. In general, models of density dependent population growth, which are roughly speaking “one hump” functions, fulfil this condition, while genetic problems, in which the selective forces depend on the gene frequencies, give rise to models with “two hump” functions (May [6]). These models cannot be treated with the tools developed in [2]. In what follows we show that Cull’s theorem remains valid without the abovementioned assumption on f. Though his theorems refer to functions f: [0, cc) -+ [0, cc) our results are applicable in this case, too. If we define then g:[O,l)+[O,l) and we see that a r(x) = x/(1 + x) and g= t.f.t-‘, model involving f is globally stable iff the one defined in terms of g is globally stable, since we have defined global stability in the open interval (0,l). We further derive a sufficient condition for global stability of models with “ two hump” functions and apply the results to a problem of frequency dependent natural selection.

MAIN RESULTS Let Q be the set of all functions conditions:

f: [0, l] -+ [0, l] satisfying

the following

(Al) f is continuous. (A2) f (0) = 0. (A3) f has exactly one fixpoint x* in (0,l). (A4)f(x)>xforO
we define f, (x) recursively fo(x)

=x3

fn(x)=f,-,(f(x)), THEOREM

by

n’0.

I

Let f E Q. If f has no maximum in (0, x*), then x* is g[obafly stable. If f has a maximum in xM E (0, x*), then x* is globah’y stable if and onb iffi(x) > x for all x E (x,,

x*).

229

STABILITY OF DISCRETE POPULATION MODELS

Proof. The first part of the theorem follows from the proof of Theorem 1 in [2]. For the proof of the second assertion let x E [ xM, x*). Then either fk (x) -C x* for all k > 1, which implies monotonic convergence to x* because of (A4), or there is a k > 1 with x

Then either fk+,(x) > x* for allj >, 1, which implies that fk+j(x) decreases monotonically toward x* because of (A4), or there exists anj > 1 with

(2) We then show fk(x)< fk+j+,(x). If j=l, this follows from the fact that x*) and the assumptionf,(x)> x if x E [xM,x*). Ifj> 1, there f/c(x) E [x,9 exists a y with fk (x) =gy < x* and f ( y ) = fk +, (x), since f is continuous. The assumption of the theorem then yields

Summarizing the above x E [x,,,, x*), then either

results,

we obtain

the following

conclusion:

If

(1) there exists a k such that fk+j( x) increases toward x* as j + cc, or (2) fk+j(x) decreases monotonically toward x* asj -+ co, or (3) there exists a subsequence (ki) of {k} such that fk,(x) increases towardx* asihco. It remains to prove convergence in the last case. If we fix E > 0, then because of (Al) we find a 0 < 6 < E such that If(x)- x*1 < E if ]x - x*] -C 8. By assumption there exists an 1> 0 such that Ifk,(x)- x*1 < S for all i > I and x E [x,, x*), which implies Ifk,+,(~)-x*(<~ for all i>l and XE x*). Now either ki + 1 = kj+ , or, because of (2) and (A4), [x,9

fk,(x)~fk,+,(x)

In any case we obtain Ifk(x)- x*J < E for all k > k, and x E [x,, x*), which is the asserted convergence. If x is in (0, xM), then because of (A4), eventually either fk(x) is in [x,9 x*1, in which case there will be convergence, or fk (x) > x*, in which case fk(x) = f(y) for some y in [x,, x*), and so there will still be convergence. Finally, if x > x*, then either fk(x) decreases monotonically toward x*, or there is a k > 1 such that fk( x) < x* and convergence follows from the considerations made above. n

230

GERD

ROSENKRANZ

We now apply Theorem 1 to population models with “two hump” functions. Let O0 be the subset of 0 containing all functions f with the following properties: (B 1) f is twice differentiable. (B2) There are points x, and x2 such that 0 < x, c x* < x2 < 1 and f’(x)>0 if O 0 for x > x,. THEOREM

Letf Proof:

2

E Cl,. Then x* is global& stable iff’(x,)f’(x*)

< 1.

First we note that (B2) and (B3) imply f’(xc>


< 0

(3)

for x E (x,, x1). Hencef’(x*)2 < 1 and x* is locally stable under the assumption of the theorem. Let z = min(x,, f(x,)). Then there exists a uniquely determined point y~[~,,x*)withf(y)=~.From(B2)weseethatf’(x) x for x E [x, , x *) according to Theorem 1. In view of the considerations above it is sufficient to demonstrate that f;(x) < 1 for x E (y, x*). Assume first x, < x*. Then f”(x) > 0 for x > x*, and f’(x) is strictly increasing in [x*,z). Thus f’(x*)
=f’(f(x))f’(x)


The case x* < x, can be treated similarly.

< 1.

If x* = x, then f;( x) < f’( x,)~ < 1, n

and the theorem is proved. AN EXAMPLE

Here we apply Theorem 2 to a genetic problem in which selective forces depend on the gene frequencies. We suppose a one locus, two allele model with relative frequencies x and 1 - x and fitnesses F(x) and 1 respectively. The change in gene frequency between one generation and the next will be Xll+l=

XnF(%) x,F(x,)+l-

X”

.

STABILITY

OF DISCRETE

POPULATION

MODELS

231

Frequency dependent selection implies F(x) > 1 for small x and F(x) < 1 for large x. In the following we will work with the special fitness function qx>

=

1+w1-

x/k)(l-

l-2a(l-x/k)x

x> ’

where 0 < k -c 1. With k = f we obtain the example considered (4) in the form (l), we obtain

in [6]. Writing

which defines a third degree polynomial with f(0) = 0, f( 1) = 1, and f( k) = k. It is easily seen that x* = k is locally stable if 0 < a < l/(1 - k). We further obtain x, = (k + 1)/3. Applying Theorem 2 to our model, we find that x* is also globally stable if +(k+k-‘+2)-k ‘< The two bounds

’

2(k-l)[l+k+2+l/k)]

are equal for k = i, in which case we also have x, = k.

REFERENCES and J. P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhauser, Boston, 1980. P. Cull, Global stability of population models, Bull. Math. Biol. 43:47 (1981). P. Collet

I. Gumowski and C. Mira, Dynamique chaotique - Transformations ponctuelles, Transitions Ordre- Desordre, Cepadues Editions, Toulouse, 1980. R. M. May, Biological populations obeying difference equations: stable points, stable cycles, and chaos, J. Theoret. Biol. 5 1: 5 11 (1975). R. M. May, Simple 261:459 (1976).

mathematical

models

with very

complicated

dynamics,

R. M. May, Bifurcations and dynamic complexity in ecological systems, Acad Sci. 316:517 (1979). A. G. Nobile, L. M. Ricciardi, and L. Sacerdote, On Gompertz growth related difference equations, Biol. Cybernet. 42:221 (1982).

Nature

Ann. N. Y. model

and