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S T A B I L I T Y ANALYSIS O F PREDATOR-PREY MODELS V I A THE LIAPUNOV METHOD

M. G a t t o S. R i n a l d i

O c t o b e r 1 9 7 5

R e s e a r c h M e m o r a n d a a r e i n f o r m a l p u b l i c a t i o n s r e l a t i n g t o o n g o i n g o r p r o j e c t e d a r e a s of research a t I I A S A . T h e v i e w s expressed a r e t h o s e of t h e a u t h o r s , a n d do n o t n e c e s s a r i l y r e f l e c t t h o s e of I I A S A .

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Stability Analysis of Predator-Prey Models Via the Liapunov Method

*

M. Gatto and S. Rinaldi

**

Abstract

As is well known from the classical applications in the electrical and mechanical sciences, energy is a suit- able Liapunov function; thus, by analogy, all energy

functions proposed in ecology are potential Liapunov functions. In this paper, a generalized Lotka-Volterra model is considered and the stability properties of its

non-trivial equilibrium are studied by means of an energy function first proposed by Volterra in the context of conservative ecosystems. The advantage of this Liapunov function with respect to the one that can be induced through linearization is also illustrated.

1. Introduction

One of the classical problems in mathematical ecology is the stability analysis of equilibria and, in particular, the deter- mination of the region of attraction associated with any

asymptotically stable equilibrium point. It is also known that the best way of obtaining an approximation of such regions is La Salle's extension of the Liapunov method [ 2 1

,

[ 4 ]

.

Nevertheless, this approach has not been very popular among ecologists, the main reason being that Liapunov functions (i.e.

functions that satisfy the conditions of the Liapunov method) are in general difficult to devise. One straightforward, but often not very effective, way of overcoming this difficulty is through linearization as shown in Section 3, while a more fruitful way consists in considering as candidates for Liapunov functions any functions that are analogous to the internal energy of the system.

This is the approach that is, for example, commonly followed by engineers in the analysis of mechanical systems or in the study of nonlinear electrical networks. The reason why the Liapunov method has not been widely used in ecology possibly resides in the

lack of a definition of an energy function in the context of eco- logical systems. One major exception is represented by the

*

Work partly supported by Centro di Teoria dei Sistemi, C.N.R., Milano, Italy. The paper has been presented at the 7th IFIP Con- ference on Optimization Techniques, Modelling and Optimization in the Service of Man, Nice, Sept. 8-13, 1975.

* *

Centro di Teoria dei Sistemi, C.N.R., Milano, Italy.

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pioneering work of Volterra and the more recent work of Kernex

[ T I who discussed the analogy between ecological and mechani-

cal systems in terms of energy. Nevertheless, these works are limited to conservative ecosystems, a case that seems to be very peculiar indeed.

The aim of this paper is to show how the energy function proposed by Volterra (from now on called Volterra function) quite often turns out to be a Liapunov function even for non- conservative ecosystems. In order to avoid complexity in notation and proofs, the only case that is dealt with in the following is the one of second order (predator-prey) systems, but the authors strongly conjecture that the results presented in this paper cou1.d be easily generalized to more complex

ecological models.

2. The Volterra Function

Consider the simple Lotka-Volterra model

= y (-c

+

dx)

where x and y are prey and predator populations and (a,b,c,d) are strictly positive constants. This system has a non-trivial

- -

equilibrium (E,y) given by (x,y) = (c/d, a/b) which is simply stable in the sense of Liapunov. Moreover, any initial state in the positive quadrant gives rise to a periodic motion.

This can easily be pivved by means of the energy function proposed by Volterra,

= (xi; - 109 X I ; ) + p(y/y

-

log y/y)

-

(1

+

p), (2)

where

since this function is constant along any trajectory and its contour lines are closed lines in the positive quadrant. In other words, the Volterra function (2) is a Liapunov function because it is positive definite and its derivative dV/dt is negative semidefinite (identically zero).

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In the following, the Volterra function will be used in relation with non-conservative ecosystems of the form:

where f and g are continuously differentiable functions. More- over, we assume that there exists a non-trivial equilibrium

> 0 and that the positive quadrant is an invariant set

for system (3) so that it can be identified from now on with the state set of the system.

3. Linearization and the Liapunov Equation

Liapunov functions can, of course, be constructed by solving the so-called Liapunov equation. This procedure is now briefly described so that the advantage of the Volterra function can be better appreciated in the next sections. Let

be the variations of

-

prey

-

and predator populations with respect to the equilibrium (x,y). Then the linearized system associ- ated with this equilibrium is given by

- - - -

where (fxIfyIgx ,gy) are the partial derivatives of f and g

- -

evaluated for (x,y) = (x,y)

.

Now, assume that the matrix F has eigenvalues with negative real parts, which implies that the equilibrium is asymptotically stable (recall that the converse is not true). Under this assumption Liapunov's equation (matrix equation)

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has one and only one solution in the unknown matrix P for any positive definite matrix Q. Moreover, the matrix P is positive definite and the function

is a Liapunov function because its derivative

is negative definite. In conclusion, the Liapunov function (6) can be very easily determined by solving equation (5) with F given as in equation (4) and with Q positive definite

(e.g. Q = identity matrix). The only limitation to the appli- cability of this method is the assumption on the eigenvalues of the matrix F: for example, the Lotka-Volterra model ( 1 ) cannot be discussed in this way, since the F matrix has purely imaginary eigenvalues. Nevertheless, even when this method can be applied, the results are not in general as satisfactory as the ones that can be obtained by means of the Volterra

function as shown in the next section.

4. The Volterra Function as a Liapunov Function

Consider the generalized Lotka-Volterra model ( 3 ) and the Volterra function V given by equation (2). Then, the derivative of the Volterra function along trajectories is given by

In order to study dV/dt in a neighborhood of the equilibrium

( x , y ) ,

it is possible to expand this function in ~ a y l o r series

up to the second order terms, i.e.

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Since

eq. (7) becomes

dv

- 1

[Ax &y]

- 2

f

Y

- + -

2; ad: d z

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T h e r e f o r e t h e s e c o n d o r d e r a p p r o x i m a t i o n o f dV/dt t u r n s o u t t o b e a homogeneous q u a d r a t i c form; by s t u d y i n g t h e n e g a t i v e o r p o s i t i v e d e f i n i t e n e s s o f s u c h a form, i t i s p o s s i b l e t o d e r i v e s u f f i c i e n t c o n d i t i o n s f o r t h e V o l t e r r a f u n c t i o n t o b e a

Liapunov f u n c t i o n . More p r e c i s e l y , by a p p l y i n g t h e well-known S y l v e s t e r c o n d i t i o n s a n d p e r f o r m i n g e a s y c o m p u t a t i o n s , w e o b t a i n

i -

2 dV p o s i t i v e d e f i n i t e

. ( l o )

d t (bqx + d z Y

)

> 4bdF g'

x Y

N o t i c e t h a t t h e s e c o n d i t i o n s a r e o n l y s u f f i c i e n t f o r L i a p u n o v methods t o b e a p p l i c a b l e ; t h u s , e v e n i f t h e s e c o n d i t i o n s a r e n o t s a t i s f i e d , i t i s p o s s i b l e t h a t t h e V o l t e r r a f u n c t i o n t u r n s o u t t o b e a L i a p u n o v f u n c t i o n ( s e e Example 2 ) .

A s f a r a s t h e s t u d y o f s t a b i l i t y p r o p e r t i e s i n t h e l a r g e i s c o n c e r n e d , t h e V o l t e r r a f u n c t i o n i s d e f i n i t e l y a d v a n t a g e o u s w i t h r e s p e c t t o t h e q u a d r a t i c f o r m s d e r i v e d by means o f

L i a p u n o v ' s e q u a t i o n ( 5 )

.

T h i s i s a p p a r e n t i n t h e c a s e o f g l o b a l s t a b i l i t y ; i n facL, g l o b a l s t a b i l i t y c a n b e i n f e r r e d by means o f V o l t e r r a f u n c t i o n , whose c o n t o u r l i n e s i n t h e s t a t e s e t a r e c l o s e d , w h i l e t h i s i s n e v e r p o s s i b l e by means o f a p o s i t i v e d e f i n i t e q u a d r a t i c form o f t h e k i n d ( 6 ) , s i n c e t h e c o n t o u r l i n e s a r e n o t c l o s e d (see Examples 1 and 2 ) . 5 . Examples

T h i s s e c t i o n i s d e v o t e d t o c l a r i f y i n g by means o f some e x a m p l e s w h a t h a s b e e n d i s c u s s e d a b o v e , w i t h p a r t i c u l a r

e m p h a s i s on t h e t r a d e - o f f s b e t w e e n t h e V o l t e r r a f u n c t i o n and t h e q u a d r a t i c L i a p u n o v f u n c t i o n t h a t c a n b e o b t a i n e d by

s o l v i n g t h e L i a p u n o v e q u a t i o n .

Example 1

The f i r s t example i s a s i m p l e s y m m e t r i c c o m p e t i t i o n model f o r two s p e c i e s d e s c r i b e d by t h e f o l l o w i n g e q u a t i o n s

(see May [ 3 1 ) :

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a =

d t y ( k 2

-

y

-

a x ) ,

where k l l k 2 a n d a a r e p o s i t i v e p a r a m e t e r s . Provided. t h a t

- -

a n o n - t r i v i a l e q u i l i b r i u m ( x , y ) e x i s t s a n d i s g i v e n by

T h u s , t h e m a t r i x F o f t h e l i n e a r i z e d s y s t e m i s g i v e n by

a n d i t s e i g e n v a l u e s h a v e n e g a t i v e r e a l p a r t s , p r o v i d e d t h a t i t s t r a c e i s s t r i c t l y n e g a t i v e and i t s d e t e r m i n a n t i s s t r i c t - l y p o s i t i v e . T h e s e c o n d i t i o n s a r e o b v i o u s l y s a t i s f i e d i f a < 1 . On t h e o t h e r h a n d , a l s o t h e s u f f i c i e n t c o n d i t i o n s g i v e n by e q . ( 9 ) work w e l l . I n f a c t

and

p r o v i d e d t h a t a < 1 .

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However, the Volterra function guarantees the global stability of the equilibrium. This can be easily under- stood when taking into account that there is no error in the Taylor expansion (7), because the functions f and g are linear. Thus, dV/dt is negative definite in the state set and global stability follows from La Salle's conditions.

Example 2

Consider the well-known modification obtained from the classical Lotka-Volterra model, when assuming, in the absence of predation, a logistic growth for the prey:

3

= y(-c

+

dx) dt

If ad > kc a non-trivial equilibrium

exists, and linearization around it yields

which has eigenvalues with negative real parts. On the other hand, it turns out that

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Therefore eq. (9) is not satisfied. Nevertheless, a direct computation yields

i.e. dV/dt is negative semidefinite. Since the locus dV/dt = 0 is not a trajectory of the system (easy to check), Krasowskyi conditions are met with and asymptotic stability can be in- ferred. Moreover, since dV/dt is negative semidefinite in the whole state set, global stability can be straightforwardly deduced.

Example 3

A third example is given to show how a subregion of the region of asymptotic stability can be found by means of the Volterra function.

Consider a situation where the prey, in the absence of predators, has an asymptotic carrying capacity B and a

minimum density a, below which successful mating cannot

overcome the death processes. This model can be described by

where a, B,y are positive parameters which are supposed to satisfy the relations

It is easy to check that there exists only one non-trivial

- -

equilibrium given by (x,y) = (C

-

a) (c

-

B ) Y

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T h i s e q u i l i b r i u m i s n o t g l o b a l l y s t a b l e , s i n c e t h e o r i g i n of t h e s t a t e s p a c e i s a l s o a s y m p t o t i c a l l y s t a b l e . The r e g i o n s A and B o f a s y m p t o t i c s t a b i l i t y o b t a i n e d by

s i m u l a t i o n f o r p a r t i c u l a r v a l u e s o f t h e p a r a m e t e r s a r e shown i n F i g u r e 1 . I t i s p o s s i b l e t o d e t e r m i n e an apyxoxi- m a t i o n o f r e g i o n A by means o f t h e V o l t e r r a f u n c t i o n . I n f a c t ,

d V - X

- -

(

-

1) ( ( x

-

a ) ( x

-

B )

-

y y )

+

y

(Y -

1) ( t - c + X )

d t x x Y

i s n e g a t i v e s e m i d e f i n i t e i n t h e r e g i o n

s i n c e ( a

+

8 ) / 2 4 c . Moreover t h e s t r a i g h t l i n e x =

X,

where dV/dt = 0, d o e s n o t c o n t a i n a n y p e r t u r b e d t r a j e c t o r y . There- f o r e t h e r e g i o n bounded by t h e c o n t o u r l i n e o f t h e V o l t e r r a f u n c t i o n t h a t i s t a n g e n t t o t h e s t r a i g h t l i n e x = a

+

B

-

c

( s e e F i g u r e 1 ) r e p r e s e n t s a n e s t i m a t e o f t h e r e g i o n of a t t r a c t i o n , s i n c e La S a l l e ' s c o n d i t i o n s a r e s a t i s f i e d .

6 . C o n c l u d i n a Remarks

The e n e r g y f u n c t i o n p r o p o s e d by V o l t e r r a h a s b e e n u s e d i n t h i s p a p e r t o a n a l y z e t h e a s y m p t o t i c b e h a v i o r o f non- c o n s e r v a t i v e e c o s y s t e m s o f t h e p r e d a t o r - p r e y t y p e . The main r e s u l t i s t h a t t h e V o l t e r r a f u n c t i o n t u r n s o u t t o b e a

w e l l - d e f i n e d L i a p u n o v f u n c t i o n f o r a l a r g e c l a s s o f s y s t e m s and t h e r e f o r e a l l o w s t h e d i s c u s s i o n o f t h e l o c a l and g l o b a l s t a b i l i t y p r o p e r t i e s o f s u c h s y s t e m s . The V o l t e r r a f u n c t i o n d e f i n i t e l y seems a d v a n t a g e o u s w i t h r e s p e c t t o t h e L i a p u n o v f u n c t i o n s t h a t c a n b e o b t a i n e d t h r o u g h l i n e a r i z a t i o n , p a r t i - c u l a r l y i n t h e c a s e o f g l o b a l s t a b i l i t y . Moreover, i t i s w o r t h w h i l e n o t i n g t h a t t h e V o l t e r r a f u n c t i o n i s a l s o o f i n t e r e s t when t h e e q u i l i b r i u m s t a t e u n d e r d i s c u s s i o n i s un- s t a b l e . The r e s u l t s o b t a i n e d i n t h i s p a p e r a l l o w u s t o p r o v e i n a v e r y s i m p l e form some g e n e r a l p r o p e r t i e s s u c h a s t h e f o l l o w i n g : i f t h e f u n c t i o n f and g i n t h e g e n e r a l model ( 3 )

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F i g u r e 1 . The e s t i m a t i o n o f t h e r e g i o n A o f a t t r a c t i o n by means o f t h e V o l t e r r a f u n c t i o n .

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are linear and satisfy eq. (9), then the local stability of an equilibrium implies its global stability.

Acknowledgement

The authors are grateful t o Dr. Dixon D. Jones for his helpful suggestions and advice.

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R e f e r e n c e s

[ I ] K e r n e r , E . H . " A S t a t i s t i c a l Mechanics o f I n t e r a c t i n g B i o l o g i c a l S p e c i e s . " B u l l . Math. B i o p h y s . , - 19

( 1 9 5 7 ) , 121-146.

[ 2 ] La S a l l e , J . P . "Some E x t e n s i o n s o f L i a p u n o v ' s Second Method." IRE T r a n s . o n C i r c u i t T h e o r y ,

-

7 ( 1 9 6 0 )

,

520-527.

[ 3 ] May, R.M. S t a b i l i t y a n d C o m p l e x i t y i n Model E c o s y s t e m s . P r i n c e t o n , N e w J e r s e y , P r i n c e t o n U n i v e r s i t y P r e s s , [ 4 ] Rosen, R . Dynamical S y s t e m T h e o r y i n ~ i o l o g y , V o l . 1 ,

~ i l e y - I n t e r s c i e n c e , 1970.

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