Forest-Pest Interaction Dynamics in Temporal and Spatial Domains

W O R K I I G P A P E R

FVREST-PEST KNTEXACI'ION DYNAMICS IN TEMPORAL AND SPATIAL DOMAINS

E.A. Samarskaya

February 1989 WP-89-16

-

l n t e r n a t ~ o n a l l n s t l t u t e for Appl~ed Sys~ems Analys~s

FOREST-PEST

INTERACIION DYNAMICS

IN

TEKPORAL

AND

SPATIAL

DOHAINS

E.A. Samarskaya

Working Papers a r e interim reports on work of the International Institute f o r Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

Preface

In this study mathematical models of forest-pest interaction dynamics in tem- p o r a l and spatial domains are developed.

A comparison of models with different types of insect feeding and competition shows t h a t p r o p e r t i e s of f o r e s t succession depend on insect feeding and competi- tive interactions within t h e species.

This study considers insect and seed spatial diffusion and t r a n s p o r t and shows t h a t t h e dispersion p a t t e r n s of t h e species should not b e ignored if a valid representation of r e a l i t y is t o b e presented. In s e v e r a l p a r t i c u l a r c a s e s traveling waves are obtained.

Parameter identification and inverse problems a r e discussed and finite- difference approximations and p r e p a r e d software f o r t h e interactive exploration of developed models are briefly described. Some numerical results a r e presented.

B.R. Doos

Leader, Environment Program

FDREST-PEfl

INTERACIlON DYNAMICS

IN

TEMF'ORAL

AND

SPATIAL DOMAINS E.A. S a m a r s k a y a

Introduction

Ecology a n d biology problems h a v e become increasingly p r e s s i n g . The only method of ecosystem r e s e a r c h i s simulation, a s e a c h ecosystem i s unique a n d no full s c a l e e x p e r i m e n t s a r e possible. A mathematical d e s c r i p t i o n of t h e e s s e n t i a l ecological problems h a s only r e c e n t l y come i n t o e x i s t e n c e .

In t h e mathematical t h e o r y of ecological communities t h e r e a r e two major sub- j e c t s :

1. t h e temporal dynamics of i n t e r a c t i n g populations, a n d 2. a s p a t i a l p a t t e r n of t h e community.

Historically, t h e s e s u b j e c t s h a v e been developed independently a n d , b e c a u s e of t h e mathematical difficulties, t h e majority of mathematical ecology models t r e a t only t e m p o r a l dynamics.

A s p a t i a l study of population dynamics began only r e c e n t l y . A study of t h e re- l a t i o n s between s t r u c t u r e a n d population dynamics i s of c r i t i c a l i m p o r t a n c e , both to o u r g e n e r a l understanding of t h e b e h a v i o r of t h e ecosystem a n d to our a b i l i t y to manage s u c h systems effectively.

The r e m a r k s of Okubo (1980) in h i s book, -&on a n d Ecological Problems:

Mathematical Models, m e r i t s consideration:

"It may b e optimistic, but I f e e l t h a t through t r i a l and e r r o r t h e use of mathematical models in t h e field of ecological diffusion will eventually lead t o t h e establishment of laws and basic equations."

A mathematical treatment is indispensable if t h e dynamics of ecosystems are to b e analyzed and predicted quantitatively. This f a c t i s becoming m o r e widely ac- c e p t e d (see, f o r example, Pielou, 1977; Clark, 1979; Levin, 1979,1981; Okubo, 1980;

Hallarn and Levfn, 1986; Svfrezhev, 1987).

When t h e model r e p r e s e n t s t h e ecosystem a c c u r a t e l y , t h e n a n important as- p e c t of t h e modelling i s control of t h e biological system. A study of insect-forest systems is considered n e c e s s a r y f o r predicting f o r e s t dynamics and p e s t manage- ment (Bell, 1975; Holling e t al., 1975; Holling and Dantzig, 1977).

Long-term relations between f o r e s t r e s i s t a n c e and pest population cannot be described p r o p e r l y without consideration of spatial dynamics. I t i s difficult t o o v e r s t a t e t h e necessity of taking into account t h e r o l e of s p a t i a l heterogeneity where pest management i s concerned. Even t h e best of long-term studies of local population dynamics fail t o make s e n s e in t h e absence of attention t o insect disper- sal. Consideration of spatial e f f e c t s fundamentally changes o u r view of t h e organi- zation of ecological communities. Models become aids t o asking b e t t e r questions and help focus s c a r c e r e s e a r c h funds, manpower, and opportunities, where they will d o most good.

Studies of d i s p e r s a l o r spatfal heterogeneity are complex but, a t t h e same time, v e r y urgent. A specffic problem can sometimes b e solved analytically but usually one must r e l y on computer calculations. H e r e t h e computer s e r v e s as t h e only possible tool for model t r e a t m e n t (Okubo, 1980).

Effective management of t h e forest-pest system r e q u i r e s an understanding of t h e consequences of a l t e r n a t i v e management s t r a t e g i e s (Bell, 1975; Holling and Dantzig, 1977). Mathematical models of ecology give t h e possibility t o consider dif-

f e r e n t situations, t o come t o some conclusions, and t o discuss implications f o r f o r e s t - p e s t management.

Of c o u r s e , t h e main problem i s c r e a t i n g a mathematical m o d e l of t h e o b j e c t u n d e r study. An a d e q u a t e model i s half t h e s u c c e s s . I t i s n e c e s s a r y not only t o write down a l l r e l e v a n t mathematical r e l a t i o n s , but a l s o to h a v e a clear idea as to which of t h e s e r e l a t i o n s i s of p r i m a r y and s e c o n d a r y importance. The phenomenon, b r o k e n down into elementary physical p r o c e s s e s , should not lose i t s i n t e g r i t y in t h e model.

A s some a u t h o r s point o u t ( s e e , f o r example, Banks and Kareiva, 1983). be- f o r e applying models t o r e a l experimental systems i t a p p e a r s n e c e s s a r y t o t e s t t h e i r p e r f o r m a n c e a g a i n s t "data" g e n e r a t e d by equations. T h e r e f o r e , i n t e r a c t i o n s between d i f f e r e n t components of t h e system a r e studied t o determine a minimum set of n e c e s s a r y information a b o u t t h e system. So, initially, o n e should investigate t h e p r a c t i c a l issues such a s t h e amount of d a t a r e q u i r e d , t h e a c c u r a c y of t h e method, and t h e computational h a z a r d s . A l a r g e s c a l e of complexity and d e t a i l may b e n e c e s s a r y in o r d e r t o discuss t h e main c h a r a c t e r i s t i c s of t h e system.

Studying t h e e f f e c t s of temporal and s p a t i a l dynamics r e q u i r e s additional d a t a beyond t h e d a t a needed f o r temporal models. T h e r e f o r e , t h e problem of parame- ters and d a t a becomes v e r y important. I t i s u r g e n t t o study i n v e r s e problems and t o apply estimation and optimization techniques.

The main p u r p o s e s of mathematical models of ecosystems are: to s e a r c h f o r p a r t i a l solutions, t o examine limiting c a s e s , to p r o v i d e qualitative dimensional analysis, to e v a l u a t e t h e dependence of t h e solution o n v a r i o u s p a r a m e t e r s

-

whether i t i s continuous o r p r o n e t o i n c r e a s e unlimitedly, etc. Mathematical models f o r population t r y

to

d e s c r i b e t h e behavior of t h e system by using s t a b l e points, s t a b l e c y c l e s and a p p a r e n t c h a o s (May, 1976; Pielou, 1977; S v i r e z h e v , 1987). Of s p e c i a l importance are t h e implications f o r p e s t o u t b r e a k s , where "ca-

t a s t r o p h e s " , in both t h e mathematical and t h e biological s e n s e , may o c c u r (May, 1976; S v i r e z h e v , 1987).

The main goals of t h i s p a p e r are:

(i) to develop some t e m p o r a l mathematical models of i n s e c t - f o r e s t dynamics by taking i n t o a c c o u n t i n t r a s p e c i f i c competition;

(ii) to study s p a t i a l dynamics a n d h e t e r o g e n e i t y ;

(iii) to c o m p a r e models which d e s c r i b e temporal and s p a t i a l dynamics of insect- f o r e s t systems with t e m p o r a l models a n d to discuss c o n s i d e r a t i o n s of s p a c e in- fluence o n t h e systems' b e h a v i o r d e s c r i p t i o n .

Our intention in c o n s t r u c t i n g mathematical models f o r i n s e c t - f o r e s t dynamics i s t o understand t h e way in which d i f f e r e n t kinds of biological and physical in- t e r a c t i o n s a f f e c t t h e dynamics of f o r e s t and p e s t . This p a p e r will t r y t o point o u t what new information c a n b e obtained by taking into a c c o u n t d i f f e r e n t n a t u r e ef- f e c t s and by studying s p a t i a l structure-population dynamics.

In Section I, t h e temporal dynamics of t h e system are c o n s i d e r e d . In- t r a s p e c i f i c competition and c a s e s w h e r e i n s e c t s f e e d both on young a n d old t r e e s are t a k e n i n t o a c c o u n t .

In Section 11 t h e models which d e s c r i b e both t e m p o r a l and s p a t i a l dynamics with consideration to insect migration are p r e s e n t e d .

In Section 111, f o r e s t - p e s t i n t e r a c t i o n dynamics in h e t e r o g e n e o u s environ- ments i s studied.

Section IV i s devoted to t h e investigation of a model which d e s c r i b e s two-age f o r e s t dynamics with seed d i s p e r s a l .

As an a n a l y t i c a l t r e a t m e n t may b e c a r r i e d o u t only in c e r t a i n c a s e s , i t i s n e c e s s a r y to p r o v i d e a computer experiment. In Appendix A t h e finite-difference approximations. a n d in Appendix B t h e s o f t w a r e which was p r e p a r e d and used f o r numerical experiments, are b r i e f l y d e s c r i b e d .

I. Spatio-Temporal Forest-Pest Interaction Dynamics

1. Basic Model

The influence of i n s e c t p e s t s o n t h e a g e s t r u c t u r e dynamics of f o r e s t systems h a s n o t b e e n extensively studied in mathematical ecology. I n Antonovsky et al.

(1988) t h e t e m p o r a l mathematical models of two-age f o r e s t , a f f e c t e d by insect- p e s t , are c o n s i d e r e d .

H e r e u a n d v are d e n s i t i e s of "young" a n d "old" t r e e s ; N i s i n s e c t d e n s i t y , p

=

^p ( v ) i s f e r t i l i t y of t h e s p e c i e s , h = h ( v ) and s

=

s ( u ) a r e d e a t h a n d aging r a t e s , E

=

^E ( u , V

,N)

i s t h e mortality r a t e of i n s e c t s , B

=

B ( u , N , b ), C

=

C ( v , N , b ), where b i s a coefficient which r e p r e s e n t s a d e p e n d e n c e of "young" t r e e mortality o n t h e density of "old:' t r e e s .

Terms u N a n d vN r e p r e s e n t t h e i n s e c t - f o r e s t i n t e r a c t i o n , a i s a p a r a m e t e r a n d d e s c r i b e s how i n s e c t s f e e d . I n [14] a

=

⁰ a n d a

=

1 only a r e c o n s i d e r e d . When a

=

0, system ( 1 ) d e s c r i b e s t h e c a s e of i n s e c t s feeding only o n "old" t r e e s a n d when a

=

1, i n s e c t s f e e d only o n "young" trees.

In t h e s e models, cases in which i n s e c t s f e e d both o n "old" a n d "young" t r e e s are n o t c o n s i d e r e d . In t h i s p a p e r d i f f e r e n t t y p e s of i n s e c t s feeding are studied, t h e r e f o r e , let a E [ O , l ] .

Models in Antonovsky et al. (1988) d o n o t c o n s i d e r i n t r a s p e c i f i c competition.

The formation and maintenance of selfaggrandizing systems are t h e r e s u l t of ap-

p r o p r i a t e nonlinear couplings and of competition between t h e e n t i t i e s constituting t h e ecosystem. Competition becomes significant whenever t h e r e s o u r c e s n e c e s s a r y f o r s u r v i v a l of biological components are limited. T h e r e f o r e , competition i s in- cluded in t h e models. When f e r t i l i t y of t h e "old" trees means s e e d production, s e e d d i s p e r s a l i s t a k e n into consideration.

G e n e r a l "directed movement" mechanisms such as convection of s e e d s , a n d at- t r a c t i v e phenomenon in population d i s p e r s a l models, a r e t a k e n into account.

Notice t h a t t h e b a s i c model in form (1) i s obtained from t h e initial o n e by a l i n e a r c h a n g e of v a r i a b l e s . In t h i s work, models obtained by t h e c h a n g e of v a r i - a b l e s are studied.

2. General Model

Consider t h e so-called g e n e r a l model which i s obtained from t h e mass balance laws and t h e b a s i c model ( s e e Fig. 1).

-b

+ V (DZVN) - V (VN),

w h e r e t i s time, z a n d y are t h e C a r t e s i a n s p a t i a l coordinates: u

=

u ( z , y , t ), u = u ( z , y , t ) , N = N ( z , y , t ) . FunctionsD1 = D 1 ( u , z , y , t ) , D z

=

( N , z , y , t ) a r e t h e diffusion coefficients f o r seed and i n s e c t s correspondingly. In g e n e r a l c a s e s t h e y may b e determined in two d i r e c t i o n s :

D*

=

IDi" ,Dp], ²

=

1, 2.

-b

The t e r m s involving V r e p r e s e n t a g e n e r a l " d i r e c t e d movement" mechanism.

-b -b

In g e n e r a l , t h e velocity V

=

V ( 2 , y , t ) r e p r e s e n t s convective/advective movement.

The t e r m s V (Dl V p v ) a n d V (D2 V N) r e p r e s e n t s e e d a n d i n s e c t diffusion c o r r e s p o n d i n g l y .

L e t u s study system (2) in domain C l

c

R' at time t

>

0 with initial a n d boun- d a r y conditions f o r u , v ,N, 2 0 , t 2 0.

F o r studying t h e system along with c e r t a i n initial conditions

u ( 2 ,Y ,0)

=

u o ( 2 ,Y). v ( 2 ,V ,O)

=

v o ( 2 ,Y 1, N ( 2 , ~ lo)

=

No ( 2 , Y boundary conditions are c o n s i d e r e d .

2.1 Boundary Conditions

Nowadays, f o r e s t p a t c h e s e x i s t as m o r e or l e s s isolated islands s u r r o u n d e d by a g r i c u l t u r a l a n d u r b a n land (Johnson et a l . , 1981). S p a t i a l boundary conditions c a n b e s p e c i f i e d in v a r i o u s ways. Two t y p e s are studied (Okubo, 1980).

Let P b e population density (pv o r N ) . Consider t h e following conditions at t h e b o u n d a r y

a n.

a) P r e s c r i b e d population d e n s i t i e s a t t h e boundary:

~ ( z , y , t ) = ~ ( t )

a t a n .

This condition r e p r e s e n t s a population r e s e r v o i r at t h e boundary.

When a h a b i t a t i s s u r r o u n d e d b y a completely h o s t i l e environment, t h e boun- d a r y may b e t r e a t e d as a n a b s o r b i n g boundary, 1.8. population density i s e q u a l to z e r o at t h e boundary:

P ( z , y , t ) = O at 8n.

This condition means t h a t t h e r e is n o f o r e s t f o r i n s e c t s a n d t h e r e f o r e n o in- sects to f e e d o n trees.

b) P r e s c r i b e d flux a c r o s s t h e boundary. Immigration o r emigration a c r o s s t h e boundary may b e r e p r e s e n t e d by t h e condition

- v V P = W ( t ) ( v = D 1 o r v = D 2 ) a t

an.

Figure 2 i l l u s t r a t e s how from one f o r e s t island, a new f o r e s t island may b e ob- tained by means of s e e d t r a n s p o r t

-

seed r a i n around a seed s o u r c e . S o if t h e r e is a n o u t b r e a k of insects on one f o r e s t island, i t may c a u s e insect o u t b r e a k s on o t h e r f o r e s t islands. T h e r e i s insect immigration f o r t h e second island and emigration f o r t h e f i r s t island.

When a h a b i t a t boundary i s completely closed t o t h e population

-

t h a t means a fenced population. I t s flux c a n b e considered t o b e z e r o a c r o s s t h e boundary (so- called reflecting boundary)

W(t) = O a t

an.

This equation states t h a t no flux of population o c c u r s a c r o s s t h e domain.

2.2 D i f f e r e n t Types o f hsect Diffusion C o e f f i c i e n t

In a c c o r d a n c e with n a t u r e ' s p r o c e s s e s , different t y p e s of insect diffusion may b e considered.

(i) Isotropic diffusion with a constant diffusivity: D2

=

c o n s t .

(ii) The diffusivity i s a function of trees ("old" and "young") densities:

D2 = D o [alpha u

+

⁽¹

-

^{a )u]. Do}

=

const.

Many s p e c i e s of insects make use of smell, so a t t r a c t i v e diffusion c a n b e con- sidered. The diffusivity i n c r e a s e s with t h e density of tree i n c r e a s e .

(iii) The diffusivity i s a function of insect density:

f o r example,

T h e r e f o r e t h e diffusivity i s high d u e to t h e high density of insects.

(iv) The combination of cases (ii) a n d (iii):

2.3 Different Typea ^of Inrect D e a t h R a t e

Different t y p e s of i n s e c t mortality are studied. Some important a s p e c t s of in- sect d e a t h c o n c e r n t h e following:

(i) Death by n a t u r a l c a u s e s only means t h a t E

=

cO

=

c o n s t .

(ii) Death by i n t r a s p e c i f i c competition a ) E

=

c0 N , EO

=

c o n s t , t h e r e f o r e

b ) competition d e p e n d s on feeding p a t t e r n s :

E

=

^{E O N} t h a t means, a u + ( I - a ) v

In t h i s p a r t i c u l a r c a s e , mortality i s high d u e to low tree density.

(111) Death by n a t u r a l and i n t r a s p e c i f i c competition c a u s e s ;

From a biological point of view, a l l of t h e s e c a s e s mean considering d i f f e r e n t a s p e c t s of n a t u r e ' s p r o c e s s e s . From a mathematical point of view, d i f f e r e n t t y p e s of diffusion equations ( l i n e a r a n d nonlinear) are studied.

11. Temporal Dynamica

1. Consider a n i n s e c t population which i s closed t o migration. Ignoring t h e e f f e c t s of s p a c e , a n d t h e r e f o r e of wind a n d diffusion, w e o b t a i n system (1) from system (2). L e t u s study d i f f e r e n t t y p e s of i n s e c t mortality, t h e case when a ^E [0,1]

and a l l p a r a m e t e r s are c o n s t a n t .

The main p u r p o s e of t h i s study i s to find o u t how t h e consideration of d i f f e r e n t n a t u r e p r o c e s s e s influence t h e solution. Consider t h e following d i f f e r e n t models:

Model A:

I n s e c t

-

f o r e s t dynamics with i n s e c t d e a t h by n a t u r a l c a u s e s only:

E

=

c0

=

c o n s t . When a

=

0 a n d a

=

I t h e b a s i c model i s obtained.

Model B:

I n s e c t

-

f o r e s t dynamics with i n s e c t d e a t h by i n t r a s p e c i f i c competition, without consideration of dependence on tree density ( s e e (a) from (ii)).

Model C:

I n s e c t

-

f o r e s t dynamics with i n s e c t d e a t h by i n t r a s p e c i f i c competition, which i s d e p e n d e n t o n tree density ( c a s e (b) from (ii)).

2. The s t a b i l i t y of t h r e e models i s studied. The main i n t e r e s t i s n o t in t h e a l g e b r a i c d e t a i l s b u t in t h e following questions: which f a c t o r s d e t e r m i n e t h e number of equilibrium points; w i l l t h e system t r a c k environmental v a r i a t i o n s o r will i t a v e r a g e o v e r them; which q u a n t i t i e s in t h e equations are biologically signifi- c a n t ?

Models A , B , and C h a v e d i f f e r e n t numbers of equilibrium points but a l l of them h a v e t h e same points as t h e model considered in Antonovsky et a l . (1988). The o r i - gin E o = (0,0,0) i s always a n equilibrium, i t h a s no biological significance. On t h e i n v a r i a n t p l a n e N

=

^0, t h e r e may e x i s t e i t h e r o n e o r two equilibria with nonzero coordinates.

Table 1 il l u s t r a t e s t h e maximum possible number of equilibrium points f o r t h e d i f f e r e n t models. The number of t h e s e points depends o n t h e o r d e r of t h e c o r r e s p o n d i n g a l g e b r a i c equation.

Table 1.

I I

M o d e l

1

^{a = O}

1

^{a = l}

I

O < a < l ,

Analytically a n d numerically, t h e r e l a t i o n s h i p between t h e solution b e h a v i o r , t h e number of equilibrium points, a n d t h e t y p e of i n s e c t d e a t h are obtained.

The r e s u l t s of d i f f e r e n t models, numerically i n t e g r a t e d by a computer, are p r e s e n t e d in F i g u r e s 3-7. The pamimeters are chosen in a c c o r d a n c e with Antonov- s k y et al. (1988). T h e r e are obvious q u a l i t a t i v e d i f f e r e n c e s between t h e computed solutions f o r d i f f e r e n t models. From Figures 3, 4, 5, a n d 6, i t i s e a s y to see how t h e e f f e c t of within-population competition influences t h e solution. From Figures 7 a n d 8, o n e c a n see how t h e s t r u c t u r e of t h e solution depends on t h e varying of

coefficient a. For different t y p e s of insect feeding (i.e., different values of a ) dif- f e r e n t types of solutions are obtained. In Figures 7 and 8, t h e r e s u l t s of Model C a r e p r e s e n t e d . All t h e s e figures i l l u s t r a t e t h a t p r o p e r t i e s of f o r e s t succession depend on competitive interactions within and between species.

T h e r e f o r e , d i f f e r e n t t y p e s of nonlinearities completely change t h e behavior of a system. But t h e s e d i f f e r e n t t y p e s of nonlinearities a p p e a r from complicating by a consideration of t h e n a t u r a l world processes. This analysis of t h e models shows how n e c e s s a r y i t i s to t a k e into account t h e physical c h a r a c t e r i s t i c s of t h e medium, without which t h e model would b e useless

-

obtained r e s u l t s would not b e a valid r e p r e s e n t a t i o n of r e a l i t y .

Future s t e p s of p r e s e n t e d model development a r e t o study t h e case of parame- t e r dependence on densities of t r e e s and insects and t o discuss how i t e f f e c t s t h e solution of t h e models.

3. Consider t h e stability of Model B f o r a p a r t i c u l a r c a s e a

=

1. This means t h e f o r e s t - p e s t ecosystem with intraspecific competition ( E

=

E O N ) , when insects feed on "young" trees only. From Table 1, one obtains evidence t h a t t h e r e may ex- i s t from one to five equilibria in t h e f i r s t o c t a n t

R:

:

Eo

=

( 0 , 0 , 0 ) , E l

=

( U 1 , ~ 1 , 0 ) , E 2

=

( u 2 , v 2 , 0 ) , E3,4

=

( U 3 , 4 ' V 3 , 4 * 4 , 4 1 1

where

Equilibrium a p p e a r in system (1) o n t h e line L l

=

I ( p , h ) , P

=

s h j

On t h e line

equilibrium

E2

coalesces with equilibrium Eo a n d d i s a p p e a r s from

R:.

Equilibrium E4,5 a p p e a r s in system ( 1 ) o n t h e l i n e

On t h e line L Z (if q

<

2) equilibrium E 4 coalesces with equilibrium E o and d i s a p p e a r s f r o m

R Q .

If q

=

2, t h e r e e x i s t s only equilibrium E g , which c o a l e s c e s with equilibrium E o on t h e line L l and d i s a p p e a r s from R:

.

If q

>

2, t h e r e e x i s t equilibrium E g only when p

>

( s

+

1 ) h . On t h e line L i t coalesces with Eo. T h e r e - f o r e , t h e p a r a m e t r i c p o r t r a i t of Model B d i f f e r s from t h e c o r r e s p o n d i n g p o r t r a i t of t h e model d e s c r i b e d in Antonovsky et al. ( 1 9 8 8 ) . By means of l i n e a r s t a b i l i t y t h e o r y p a r a m e t r i c conditions a r e obtained.

In F i g u r e 9, t h e solution numerically i n t e g r a t e d by t h e c o m p u t e r i s p r e s e n t e d f o r a p a r t i c u l a r case ( a

=

1) of Model B. Analytically, i t i s obtained t h a t on line

L 5 =

t ( p , s ) : P

=

sq

+

s h j

e x i s t s equilibrium

E,

=

( h + q , l , q ) ,

stable when certain conditions on t h e parameters t a k e place.

IIL

Forest-Pd Intmction Dynamics in Heterogeneou Environments

A c o n s i d e r a t i o n of s p a t i a l e f f e c t s may fundamentally c h a n g e o u r view of t h e o r g a n i z a t i o n of t h e f o r e s t - p e s t system.

F o r t h e s a k e of simplicity, c o n s i d e r t h e one-dimensional diffusion-reaction system (2) with c o n s t a n t c o e f f i c i e n t s when non-diffusive t e r m s are n o t included.

This system p r o v i d e s d i f f e r e n t models as submodels a n d t h e most convenient start- ing point f o r a discussion of a mathematical modeling.

Let us study t h e following submodel of t h e spatio-temporal model (2). Consider f o r e s t - p e s t i n t e r a c t i o n dynamics in t h e one-dimensional domain O

= [OJ]

( s e e Fig.

1 0 ) .

The main i n t e n t of t h i s study is to c o n s i d e r t h e e f f e c t s of diffusion ( H a l l a m and Levin, 1986). L e t

D 2 =

c o n s t , a

=

1. System (3) h a s s p a t i a l l y uniform equilibrium.

Note (U.,V.,N ,) i s o n e of them. To study i t s s t a b i l i t y with r e s p e c t to small p e r t u r -

IV .

.

bations l e t u

=

u

. ⁺

#^{u , v}

⁼

^{v .}

⁺

^{v ,}

^-

^N

⁼

^N.

⁺

N a n d d i s c a r d h i g h e r - o r d e r t e r m s to o b t a i n t h e linearized system:

Consideration of d i s t u r b a n c e s p r o p o r t i o n a l to eih g i v e s t h e following results:

If F

< ^F,

t h e n diffusion d o e s n o t destabilize s t a b l e equilibrium.