Biology Vol. SO, No. I, pp. 67-75,
STABILITY OF DISCRETE POPULATION MODELS H
Press plc Biology
Department of Computer Science, Oregon State University, Corvallis, OR 97331, U.S.A. We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is - 1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.
1. Introduction. In the theoretical study of population growth and regulation, one-dimensional difference equation models are often used. These seemingly simple models can have quite complex behavior (May, 1974, 1976). This complicated behavior seems to arise when the reproductive rate is high. For low reproductive rates the population trajectories tend to approach an equilibrium point. Do the trajectories approach equilibrium for all starting points or only for starting points near the equilibrium? Most studies calculate conditions on the parameters which give local stability. The situation when the derivative at the equilibrium is - 1 is often ignored. We give a new method for dealing with this ;ase. Global stability seems to be harder to demonstrate than local stability. The traditional method for demonstrating global stability is the construction of a Liapunov function (LaSalle, 1976). Liapunov functions are constructed for specific population models in Fisher et al. (1979) and Goh (1979). Since onedimensional models are so special, simpler methods might exist. Simple methods are developed in Cull (1981,1986) and Rosenkranz (1983). We give several of these methods and show that these methods apply to the usual population models from the literature. Interestingly we conclude that for these usual models local and global stability coincide. Does local stability implies global stability follow from the one-humped shape of population models? Unfortunately the answer is no. Figures 1 and 2 give one-humped curves which are locally but not globally stable. Why does local imply global for the usual models? As far as we can tell this is because the usual models have a very simple analytic form. We show that when we can 67
identify a polynomial in a usual model, we will still have local implies global if the degree of the polynomial is increased by 1, but we can construct a locally but not globally stable model when the degree is increased by 2. Definitions. A population model has the form x t+1=f(x,)
where f is a continuous function from the nonnegative reals to the nonnegative reals, and there is a positive number X, the equilibrium point, so that f(O)=0
>x for OX
and, iff’(x,) = 0 and x, I X, thenf’(x) > 0 for 0 IX x, such that f (x) > 0.
We will allow the possibility that f (x) = 0 for all x > 2 and therefore that f (x) is not strictly differentiable at CLOtherwise, when we assume that a derivative exists we also assume that the derivative is continuous. A population model is globally stable iff, for all x0 such that f (x0) > 0, we have lim,, coot= 2. A population model is locally stable iff, for every small enough neighborhood of X, if x0 is in this neighborhood then x, is in this neighborhood and lim,, mx, = X. It is not immediate that global stability implies local stability but this is an easy consequence of Theorem 2 (below). 2. Conditionsfor
Assuming thatfis differentiable at 2, there is local stability if (f(Z)1 < 1 and 1ocal instability if If(Z)( > 1. Iff’(%) = 1, then f(x+s)=Z+s+O(s2) b u t from the definition of population model f (x) < x for x >X, so X+&>f (Z+e)>% for small enough positive E and similarly X-E
STABILITY OF DISCRETE ONE-DIMENSIONAL POPULATION MODELS
-1. So th e algebra needed to calculate the expansion could be overwhelming. The following continued fraction-like method should be simpler and allows one to avoid some self-inverse functions. Assume that X= 1. Define
x?-(x) - 1 (f(x)-1)(1-x)’
Then 1 will be locally stable forf(x) withy(x) = - 1, iff Pcf( 1- E))> P( 1 -a) for all small enough positive E. Let Pi be the ith coefficient in the expansion of P. From the inequality, POcan have no effect on local stability. If the first nonzero coefficient beyond PO has an odd subscript, say P2,+ I #O, then we get local stability if P2i+l > 0 and local instability if Pzi+ 1c 0. The situation is more complicated if the first nonzero coefficient is Pzi because we then also have to take account OfPzi + 1. This method.will not work if P, = - 1. This difficulty can be removed by conjugation, that is, defining a new function with the same convergence properties asf but with P,, # - 1. This may be done, for example, by using g(x) = cf(x ‘/‘)I ‘. Notice that PO+1
So if f has PO+ 1 = 0, we can choose, for instance, c = 2, so that g will have P,, + 1= $. We summarize these results in the following theorem. Stability). The population model x,, 1=f(x,) withf(l)= has 1 as a locally stable equilibrium point if
Dejke P(x) =
xf(x) - 1 (f(x)-1)(1-x)
= Po+PI(x-1)+P,(x-1)2+~~~~ P Zi+l’O
where P2i+ 1 is thejirst nonzero coeficient beyond P,. [Conjugatef, if necessary, so that PO# - l] i(Po+l)P2i+P2i+,>O, i(Po+l)P2i+P2i+,=Oand
where Pzi is thejirst nonzero coeficient beyond P,,.
3. Conditionsfor Global Stability.
In this section we give some conditions for global stability which we will apply to specific models in the next section. THEOREM
2.A population model is globally stable iflit has no cycle of period 2.
Proofs of this theorem appear in Cull (1981,1986) and Rosenkranz (1983). This theorem is really a special case of Sarkovskii’s (1964) theorem which, among other things, states that if a continuous map has a cycle then it has a cycle of period 2. This theorem applies to a wider class than the population models we have defined. For example, the assumption of a single maximum is not essential. The two essential assumptions are that the model is continuous and cannot diverge. Two corollaries of this theorem are: COROLLARY
1.1.A population model is globally stable ifl f(f(x))>xfor
1.2. A population model is globally stable if f’(UYf(x))<
1 for XC Lx,, 2).
These corollaries apply to more general models if the inequalities hold on the larger interval (0, X). Corollary 1.l gives a computational method for checking for global stability: plot the second iterate of the function and see if it is above the line y = x. This method has, of course, the usual computational problem of approximating a curve by a finite number of points and hoping that no important feature was missed by the chosen points. If the function f is relatively simple, this computational problem should not be significant. To carry out this computation, values for all of the parameters must be specified. We are interested in investigating whether local stability implies global stability and we are, thus, interested in an infinite set of parameter values, rather than one setting of the parameters. As we will see, in most population models, we can take the ‘limit’ along one parameter and fix a value for the parameter. So in a one parameter model we would be left with a situation in which we could apply the computational test, but with several parameter models it may not be possible to reduce our investigation to a model with all parameters specified. Corollary 1.2 is an example of a sufficient condition which may be easier to test than the condition of the theorem. Rosenkranz (1983) gives an example for which he shows algebraically that this sufficient condition holds. We next give two more sufficient conditions which are easy to apply to the models of the next section. The first of these sufficient conditions implies that f’(f)f’< 1, while the second sufficient condition implies thatf’Cf)f
STABILITY OF DISCRETE ONE-DIMENSIONAL
THEOREM 3 (S~~cient
A). A ~op~~atio~modes is globally stable
if If'(Z)1 5 1 (the necessary condition farlocal stability); f”(x)
Proofs of this theorem appear in Cull (1981, 1986). This theorem assumes that the required derivatives are continuous. As Huang (1986) points out the theorem is invalid iff’(x) is not continuous. THEOREM 4 ([email protected] Condition B). A population model with f’(Z) = - 1 is globally stable if k’(x) I2 on [xm, 2) where k(x) = x/f(x); g(x) 2 0 on [x,,f(x,)] where k(x)/k’(x) = g(x) + Bx and B is a constant chosen to make g(x) nonnegative;
g’(x) S 0 on Cx,,f (x,)1; 9”(x)ZO on Cx,,f(x,)l.
A proof of this theorem appears in Cull (1986). Since condition B only applies whenf’(l)= - 1, we want to relate models withy(%) > - 1 to models withy(%) = - 1. Models often have a parameter P so that the derivative at X is a decreasing function of P, and the whole function becomes steeper as P increases. The following lemma says when we can prove global stability for a range of parameter values by proving global stability for one parameter value. LEMMA. Ifthere is a ~arumeter P so that
and if x, + f =f (x,) is globally stable weep P = P,, then the model is also globally stable for all PI POsuch that the model is a population model.
The reproductive rate usually satisfies the conditions of this lemma, so we can restrict our attention to those values of the reproductive rate which give s(Z) = - 1. When the reproductive rate goes to 0, models usually degenerate to f(x) = x which is not a population model according to our definition. Of course, this degenerate model is not globally stable because each point is a neutrally stable equilibrium point. 4. Models. In this section, we will give several models from the literature (Table I). For each of these models the set of parameters which gives local stability is the same as the set of parameters which gives global stability. Most
Parameters giving global stability
Sufficient conditions satisfied
x exp(r( 1 - x))
Moran (1950) Ricker (1954) Smith (1974) May (1974) Fisher et al. (1979)
Smith (1968) May (1976)
Nobile et al. (1982)
(1 + a exp(b))x
(I+ a exp(bx))
(l+a)bx -(1 +ax)
Pennycuick et al. (1968)
5. ~i~p~~~t hotels which are Locally but not ~~o~a~~y~tu~~e. It is easy to draw population models which are locally but not globally stable by taking a stable model and leaving the curve alone near 2, but then either making the peak much higher or making the tail descend much more quickly. We have seen in the last section that for the usual models local and global stability coincide. This is due to these models being in some sense simple. We can then ask how much more complicated a model needs to be to allow local without global stability. If we restrict functions to polynomials then degree is a reasonable measure of the complexity. We will consider generalizations of models I and II and show that increasing the degree of the polynomial by 1 still
STABILITY OF DISCRETE ONE-DIMENSIONAL POPULATION MODELS
causes local and global stability to coincide, but increasing the degree of the polynomial by 2 allows local stability without global stability. LEMMA.
Any population model of the form X ,+l=x[l-a(x-l)+b(x-l)*]
is globally stable if it is luca~~ystable.
Proof. Local stability requires a I 2. Since a satisfies the parameter lemma we can assume that a = 2. We can assume that b 2 0 because otherwise there will be a second equilibrium point above x = 1. We can also assume that - 2 + b(x - 1) ~0 on (0,l) because otherwise the condition f (x) > x on (0,l) will be violated. Such a model is globally stable because it satisfies condition B. 2(1-b(x-1)) “= Cl-2(x_l)+b(x_I)2]2’2
because the denominator is greater than 1 from the assumption that - 2 + b(x - 1) < 0, and the numerator is less than 2 because b and x - 1 are both negative. Iff has a maximum x, in (0, l), then, by the definition of population model, f’x, but this implies that k’>O for x)x,,,. So I-b(x-l)>O for X)X,. The other parts of condition B are now easily verified. n If we now consider polynomial population models of higher degree, local stability without global stability is possible. A simple example is f(x)= x(x-3/2)[--2-(x-1)-6(x-l)*]. It is easy to check thatf’(l)= -l/2, so x = 1 is a locally stable equ~ibrium point. But f(I/2) = 3/2 and f (312) = 0, so starting at l/2 the population crashes to 0 in two steps and then stays at 0, and this model is not globally stable. Figure 1 gives a plot of this function. Similar arguments can be used to show the following: LEMMA.
Any population model of
where g(x) is a polynomial of degree 2, is globally stable if it is locally stable.
Again if we increase the degree of the polynomial we can get local without global stability. For example, f(x) = x eBCX)with g(x) = - 1.9(x - 1)+ (7.6 8 ln3) (x- 1)3 is locally but not globally stable because f’(l)= -0.9, but f(1/2)= 3/2 and f(3/2)= l/2. A picture of this model appears in Fig. 2.
Figure 1. A 4th degree polynomial model which is locally but not globally stable.
Figure 2. A locally but not globally stable model of the form&)=x is a 3rd degree polynomial.
These models are buffered in the sense that increasing the degree of the polynomial by 1 still gives local implies global, but increasing the degree of the polynomial by 2 allows local without global stability. Similar buffering seems to occur in the other usual models. For example, we could consider model III to be x times a polynomial in In X, and we could consider model VI to have as its denominator a polynomial raised to the b power. In all of these cases increasing the degree of the polynomial by 1 still leaves local implies global but increasing the degree of the polynomial by 2 allows local without global stability. 6. Discussion. We have shown that local and global stability coincide for the usual population models, but local and global stability do not coincide for more complicated models. Since we used two different sufficient conditions and
we can show that some of the usual models are not conjugates of some other models, we do not feel that we have a characterization of the usual models. All that we can say is that the usual models seem to be simple. We would like to have a good characterization of simple population models. We should also mention that models without global stability can exhibit very counter-intuitive behavior. For example, moving the population toward the equilibrium can put the population on a trajectory which never goes to the equilibrium. Further, very small changes can result in completely different eventual behavior. For example, a population which will converge toward equilibrium is arbitrarily close to a population which will oscillate and not converge. In a word, models which are not globally stable are not good models to use for predictions.
LITERATURE Cull, P. 1981. “Global Stability of Population Models.” Bull. math. Biol. 43, 47-58. 1986. “Local and Global Stability for Population Models.” Biol. Cybern. 54, 141-149. Fisher, M. E., B. S. Goh and T. L. Vincent. 1979. “Some Stability Conditions for Discrete-time Single Species Models.” Bull. math. Biol. 41, 861-875. Goh, B. S. 1979. Management and Analysis of Biological Populations. New York: Elsevier. Hassell, M. P. 1974. “Density Dependence in Single Species Populations.” J. anim. Ecol. 44,
Huang, Y. N. 1986. “A Counterexample For P. Cull’s Theorem.” Kexue Tongbao 31,1002-1003. LaSalle, J. P. 1976. The Stability of Dynamical Systems. Philadelphia: SIAM. May, R. M. 1974. “Biological Populations with Nonoverlapping Generations: Stable Points, Stable Cycles, and Chaos.” Science M&645-647. -. 1976. “Simple Mathematical Models with Very Complicated Dynamics.” Nature 261, 459-467.
Moran, P. A. P. 1950. “Some Remarks on Animal Populations
Dynamics.” Biometrics 6,
Nobile, A. G., L. M. Ricciardi and L. Sacerdote. 1982. “On Gompertz Growth Model and Related Difference Equations.” Biol. Cybern. 42,221-229. Pennycuick, C. J., R. M. Compton and L. Beckingham. 1968. “A Computer Model for Simulating the Growth of a Population, or of Two Interacting Populations.” J. theor. Biol. l&316-329.
Ricker, W. E. 1954. “Stock and Recruitment.” .J. Fish. Res. Bd. Can. 11, 559-623. Rosenkranz, G. 1983. “On Global Stability of Discrete Population Models.” Math. Biosc. 64, 227-23 1.
Sarkovskii, A. 1964. “Coexistence of Cycles of a Continuous Map of a Line to Itself.” Ukr. Mat. Z. 16,61-71. Smith, J. M. 1968. Mathematical Ideas in Biology. Cambridge: Cambridge University Press. -. 1974. Models in Ecology. Cambridge: Cambridge University Press. Utida, S. 1957. “Population Fluctuation, an Experimental and Theoretical Approach.” Cold Spring Harbor Symp. quant. Biol. 22, 139-151.
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