Qualitative analysis of generalized volterra models

Ecological Modelling, 17 (1982) 91-106 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

91

Q U A L I T A T I V E ANALYSIS O F G E N E R A L I Z E D V O L T E R R A MODELS

KARLHEINZ STEINM(JLLER and EBERHARD MATTH,~US Central Institute for Cybernetics and Information Processes, Academy of Sciences, Berlin (G.D.R.)

ABSTRACT SteinmiJller, K. and Matth~ius, E., 1982. Qualitative analysis of generalized Volterra models. Ecol. Modelling, 17: 91-106. Generalized Volterra models are analyzed qualitatively to provide some information concerning their applicability as ecological-modelestimators. For the general two-population system (including prey-predator, competitive, and other cases), the stability of the equilibrium points and other properties of the phase representation are investigated. For n-population systems, stability analysis is carried out mainly by means of several Liapunov functions. The persistence (or global stability of nontrivial equilibrium) of n-population systems is investigated, leading to an interaction-exclusionprinciple which implies restrictions on the interaction structure of the system, especially in the case of trophic systems.

INTRODUCTION The modelling of ecosystems has become a social necessity since traditional contra-symptomatic methods of control and management have proved inadequate for complex ecosystems. Causal understanding of system dynamics is a prerequisite for the derivation of appropriate control devices. However, the a m o u n t of insight gained and the ultimate success of the modelling effort depend strongly on the model representation used for a specific ecosystem. Therefore, a close acquaintance with all the specific static and dynamic features of the model is necessary. Up to now there has existed no theory describing what kinds of models are best suited to what kinds of ecological problems, and there is not even, as a first step toward such a theory, a comprehensive and detailed classification of ecological models. The scope, merits and limitations of various model types in ecology have not yet been investigated systematically. Ecosystem modelling is still an art, not a science. 0304-3800/82/0000-0000/$02.75

© 1982 Elsevier Scientific Publishing Company

92 Of course, the present paper is not intended to fill this vast research gap. Nevertheless, it is aimed at a qualitative analysis of the most fundamental properties of generalized Volterra systems (i.e., Volterra systems with no restrictions on parameter signs) and, by means of this, at an elucidation of some limitations and virtues of this small class of models. Whereas simulation studies are always confined to special parameter values, a theoretical analysis of the qualitative features of a model (or an appropriate, simplified metamodel) can provide a survey of possible modes of behaviour of the m o d e l - - n o t only for special parameter values but for classes of parameter constellations (e.g., constellations of parameter signs). The most simple approach to ecological modelling is of course to use linear representations (the classical, compartment-oriented approach), for example in the form of linear regression models for static problems. Obviously, any system can be approximated linearly in the vicinity of an equilibrium point, and each equilibrium point of a (generally nonlinear) system possesses its own specific linear model. As long as only small perturbations of the equilibrium points are considered, linear approximations suffice. They are of no great use when any of the following problems has to be studied: large deviations of the state variables from the equilibrium values; the existence of more than one equilibrium point and of transitions between different equilibria (and consequently, distinction between local and global stability); limit cyclic behaviour (i.e., stable oscillations). Ecosystems are known to show all of the following three features: large fluctuations (as perturbations, even in deterministic models) of population numbers due to environmental changes Or to intrinsic feedback break-ups; multiple equilibria due to the preponderance of one or another intrinsic control mechanism, and sharp shifts from one of these equilibrium points to another (see for example, Isaev et al., 1979); oscillatory behaviour of population numbers, due for example to trophic feedback. The use of linear models outside of their validity domain results in a rapid decrease of the level of significance, which can be interpreted as uncertainty due to oversimplified modelling. Other sources of uncertainty in ecological models, e.g., uncertainty of environment, variability of biological parameters, etc., are not considered here (see O'Neill and Gardner, 1979). Patten (1978) advocated an intrinsic linearity of all ecosystems, considering nonlinearities to arise from insufficient resolution of the state variables which could be avoided, for example, by the inclusion of a greater number of environmental forcing-variables. It may be that this is in principle correct, but this point of

93 view is neither constructive nor pertinent for modelling aimed at gaining insight into ecosystem behaviour. In his pioneering work, Volterra (1931) introduced "bilinear" models for prey-predator interactions. Oscillations in trophic systems and competitive exclusion (Gause, 1934) were explicable by these models. Quite generally, the main virtue of Volterra models is that they are the most simple models which reasonably map interspecies interactions. As is the case for any other autonomous model type, they are open to the inclusion of environmentally dependent forcing terms, mainly into c; as done for example by Dubois (1979), but this is usually avoided and such forced systems are hardly analytically tractable. Similarly, we do not consider here spatial relations or time lags. The principal growth and interaction differential equations for Volterra population dynamics are of the form

.Kf~:NiFi(N)=Ni(,i+~aijNj} \

/

(i,j:

1 ..... n)

(1)

J

where N = (NI,... , N,) is the vector of abundances (densities) of individuals in populations 1 ...n, ¢; is the intrinsic growth rate of population i (positive only for producers), and the interactions between the populations are mapped by the community matrix A = (aiy } . The following interaction types may be distinguished: a,i , density feedback within one population, which is in most cases negative (Verhulst term), although sometimes positive (Allee effect); aiyaj, < 0, trophic feedback, where population i is prey if aij < 0, otherwise predator; conservative trophic feedback can be described by a i / = - a j; using a suitable renormalization; a ; / < 0, aj~ < 0, an antagonistic relation (competition) between populations i and j; and a ; / > 0, a j; > 0, a mutualistic relation (e.g., symbiosis) between populations i and j. In the following, we give a survey of possible modes of behaviour of two-population Volterra systems and of the equilibria of n-population Volterra systems. We then introduce and compare some Liapunov functions for these systems, and finally investigate problems of persistence. TWO-POPULATION VOLTERRA MODELS Commonly, only special cases of two-population Volterra models are investigated, e.g., predator-prey or competitive systems, which are distinguished by a particular sign-constellation of the community matrix. In contrast, this section is devoted to the study of the general two-population Volterra system and the various modes of behaviour such a system can

94 display. The model includes intraspecies interaction terms as interspecies ones (al2, a21):

(a ll,

a22) as well

Nl = NIF,(N,, N2) = Nx(e, + a,,N, + a,2N2)

(2)

N2 = u2r:( u,, U~) = U~(,2 + a2,U, + a~U2) Putting the right-hand side of (2) equal to zero, we obtain equilibrium values E = (El, E2} for the population densities. Four equilibria result:

E ' = ( 0 , O) E 2 = (0, - ~ 2 / a ~ } E3= {-,1/a1,, 0}

(3)

E 4 = {ql, q2} with ql

=

[ a'2¢1 [/[A[, q2 = --[ alle~ [/]A[, [AI = [ allal2 [ a22£2

a21~ 2

a21a22

These equilibria are called feasible if their components are finite and nonnegative. The stability properties of the four equilibria are most important for the long-term behaviour of two-population Volterra models. By linearizing (2) at the equilibria, it is possible to calculate the Jacobian Jij(E) = ON~F~(N,, N2)/ONyl E

(4)

and its eigenvalues, which are the solutions of IJ(E)-XI[ =0

(5)

The eigenvalues A = { ~ , ?~2} for the four equilibria are obtained as A 1 = {c,, %} A 2 = {--[Al'ql/a22, --e2} A 3 = { - e l , - [ A I.q2/a,,}

(6)

A4= {T+ ¢ ~ - - D , T--~TS--D } with T = (allq~ + a22q2)/2 a n d D = IAI. qlq2. According to Liapunov's second method, an equilibrium is locally asymptotically stable if and only if all of its eigenvalues have negative real parts; it is unstable, if and only if all of its eigenvalues have positive real parts; otherwise, it is a saddle point. If the eigenvalues have zero imaginary parts, the equilibrium is a node; otherwise, a focus. Take for example E 4 = {ql, q2} : for D < 0, the two eigenvalues ~41 = T + T ~ - D and X42 = T - - ( T 2 - - D possess real parts with opposite signs, therefore E 4 is a saddle point;

95 TABLE I Stability of E 4 for D > 0

T 2 -- D 9 0 T z - D <0

T<0

T=0

T>0

stable n o d e stable focus

saddle point neutrally stable c e n t r e

unstable node u n s t a b l e focus

for D = 0 with )~41 = 2T and )k42 = 0, E 4 is a multisingular point with one (un)stable and one saddle-point sector; for D > 0 the six cases of Table I must be distinguished. In the two-population model given by (2), all information concerning the system behaviour is contained in the two-dimensional phase representation. Of course, the N~ and N 2 axes are separatrices of the system (2). Therefore, investigations may be confined to the positive (i.e., biologically feasible) quadrant. An analysis of all possible modes of behaviour of two-population Volterra systems will thus include mainly (a) an analysis of the locations of the above four equilibria, and (b) an analysis of the stability properties of these equilibria. The existence of separatrices and the general shape of the domains of attraction can in most cases be derived easily from the stability properties of the equilibria. Table II gives a survey of the possible combinations of stability properties T A B L E II Survey o f stability c o m b i n a t i o n s for feasible equilibria

IAI>0 c ~< 0 ¢2 < 0

c 1• c 2 < 0

c i >0 c2 >0

Case 1 E1 E 2, E 3 E4

IAI<0 stable saddle points unstable

Case 2 E1 stable E 2, E 3 u n s t a b l e E4 saddle point

Case 3 E l, E 2, E 3 s a d d l e p o i n t s E4 stable if T < 0 , otherwise unstable

Case 4 E 1, E 4 s a d d l e p o i n t s E2 stable if c 2 > 0 , o t h e r w i s e E3 u n s t a b l e vice versa

Case 5 EI E 2, E 3 E4

Case 6 En unstable E 2, E 3 stable E4 saddle point

unstable saddle points stable

96

in the most interesting case that all equilibria are feasible. For nonfeasible stable equilibria, extinction or unbounded growth of one or both populations is typical. The biological interpretation is in most cases obvious. In Cases 1 and 2, both intrinsic growth rates are negative and both density feedbacks all and a22 positive. Therefore, depending on the initial conditions, the populations will either become extinct (if both do so simultaneously, locally and globally stable E 1 will be attained) or will grow without any bound. In Cases 5 and 6, both intrinsic growth rates are positive. In Case 5 the populations finally attain the equilibrium E 4, which even may be globally stable because of a sufficiently large negative density feedback for both populations (see Goh, 1976). Case 6 depicts a situation of competitive exclusion. Which of the populations becomes extinct depends on whether the initial population densities are above or below the separatrix originating at E I and leading through the saddle point E 4 to infinity. The other population attains a density-feedbackcontrolled equilibrium E 2 or E 3. Case 4 also displays an extinction situation. Depending on the initial conditions, either the population having a positive intrinsic growth rate attains a density-feedback-controlled equilibrium E 2 or E 3, or the other population (having positive density feedback) grows without bound.

N2

~

N2'

,~ ~ E2~,

EI

E2

",, E3 > N2=O

NI

El

Q2 =o separatrices

ordinarytrajectories ......isoclinefor~[=0 Fig. 1. Phase portrait of two-population system: Case 3 (see text) with T < 0 . Fig. 2. Phase portrait of two-population system: Case 3 (see text) with T > 0 .

97

More complicated is Case 3, which includes the classical predator-prey systems. It splits into three subcases. For T < 0, the nontrivial equilibrium E 4 is stable, and the population densities attain it if they were initially below the separatrix leading from E 2 or E 3 (for respectively ~2 < 0 or c I < 0 ) to infinity (the domain of attraction of E4). Otherwise the population having positive density feedback grows without bound (Fig. 1). For T > 0 , E 4 is unstable, and regardless of the initial conditions the population having positive density feedback grows without bound (Fig. 2). For T = 0, E 4 is neutrally stable, and consequently oscillations around it occur below the separatrix connecting E 2 and E 3 (Fig. 3). Putting all = a22 = 0, these equilibria E 2 and E 3 a r e shifted to infinity and the classical predator-prey system of Volterra (1931) results. In this case, the entire positive quadrant is filled with neutrally stable oscillations. Transitions between the qualitatively differing patterns of Case 3 may occur as a result of continuous changes in the system parameters. Sometimes the parameters of population systems are subject to change due to for example environmental changes or control measures. For the latter in particular, it is important to know whether bifurcations take place, namely, whether there are limit cycles. Using the criterion of Dulac, Bautin (1954) proved that no limit cycles occur in two-population Volterra systems. They may occur for two-population systems having higher nonlinearities, as shown for example by Svirechev and Logofet (1978). In generalized "bilinear" Volterra systems, parameter changes lead at most to a coincidence of equilibria (e.g., E 4 coincides with E 2 for c~a22 = c2a12), resulting in sector stability of these so-called multisingular points.

12 N2=O Fig. 3. Phase portrait of two-population system: Case 3 (see text) with T----0.

98 A more complete analysis of two-population systems includes not only nonfeasible finite equilibria, but also infinite equilibria. This can be done by means of a Poincar6 transformation, e.g., by N I = 1/,/1, N 2 - = r l 2 / r l l for NI ~ oo (see Bautin and Leontovich, 1976). The new differential equation in the variables ~/1 and ~/2 has certain finite equilibria corresponding to infinite equilibrium points of the original system. For "bilinear" two-population systems, the following infinite equilibria are obtained: E s = {0, 00} E9 = {oo, aoo}

E 6 = {00,0}

E 7 : (0, - 0 0 )

r8:

{-00,0}

E ' ° - - { - oo, aoo}

with a = (all - a z l ) / ( a 2 2 - alE ). By means of the Poincar6 transformation, the whole phase representation is mapped onto a unit circle. The infinite equilibria lie on its boundary. Of course, their stability properties influence the behaviour of the system for finite values of N I and N 2 (interior of the circle). E 5 being stable implies that, at least for sufficiently large initial values, N 2 grows without any bound (see Fig. 4, corresponding to Case 4).

E5

E8

6

Fig. 4. Complete phase representation of two-population system mapped onto unit circle: Case 4 (see text).

99 N-POPULATION VOLTERRA MODELS Whereas a considerable n u m b e r of special bilinear systems containing more than two populations have been studied (e.g., three-population systems by Kerner (1961), n-population competitive systems by Rescigno and Richardson (1973) and Coste et al. (1978)), up to now a comprehensive theory of these systems has been lacking. A statistical mechanics has been developed only for very large, conservative trophic Volterra systems (Kerner, 1971). T h o u g h being in no way exhaustive, the investigations in this section are devoted to the stability of the equilibria of n-population systems, especially of the nontrivial equilibrium. The equilibria E k = {E k1,...,En} k of the Volterra model (1)

Ni=N,(~i+~aijNj)

( / = 1,...,n)

J

are the solutions of the system of equations

Eki({i+~aiyE:)=O

( i = 1,...,n)

(7,

J As any of the n equations of (7) factorizes into E~ = 0 or ,, + ~]

aijE~ = 0

J

there are altogether rn = 2" equilibria of the n-population system:

E' = (0,0 ..... o) {E ,o ..... o}

E

= {E;

, . . . , E , _ l , 0}

E m: {E~',...,Ey) The nontrivial equilibrium E m is the only equilibrium which does not lie on a hyperplane spanned by the coordinate axes. Some of these 2" equilibria may coincide; they are feasible if their components are nonnegative. As in the two-population case, the feasibility of E m is decisive for the global behaviour of the system. If E " is not feasible, at least one population is b o u n d to become extinct. Matrix calculus implies that for conservative trophic systems (atj = - a j i ) , E m is not feasible if the n u m b e r of populations n is odd (see for example, Goel et al., 1971). These systems are not persistent in the sense of the following section.

100

Using E~ = 0 for the vanishing components of the equilibria and otherwise c~ = __ Eja~jEj,k it is easy to calculate the components of the Jacobian for E~ = 0 a n d j v~ i

J,j= 0NiF~(N)/0Nj[ N=E, =

aaE~ + ,i

[ aiyEki

for E/~ = 0 a n d j = i

(8)

for E~ ~ 0

Accordingly, the eigenvalues k of J, i.e., the solutions of [J(E) - h i [ = 0, are decisive for local asymptotic stability of E. Results for qualitative stability and for the probability of stability of high-dimensional systems have been given for example by May (1973), whose most general result was that the probability of stability of a linearized ecosystem (described by J(E)) decreases with increasing complexity of the system (dimension n, or number of nonzero components of J(E)). Another important question, the existence of limit cycles, has been investigated up to now mainly numerically. Coste et al. (1978) found a limit cycle in certain special three-population competitive systems. Bifurcations of equilibrium points do not occur, because the equilibrium equations (7) factorize into linear equations, where the number of solutions does not depend on the parameters. LIAPUNOV

FUNCTIONS

FOR VOLTERRA

MODELS

Sometimes it is more easy to perform stability investigations by means of Liapunov functions (Liapunov's first method). They are especially useful when it is necessary to estimate the size of the domain of attraction of an equilibrium. Unfortunately, they are commonly difficult to calculate and there is generally a lack of good estimators. Introducing x i = N i - E i, Liapunov functions are defined as follows. A function V(x) is called a Liapunov function for the equilibrium x -- 0, if and only if there exists a region R containing x = 0 and: V is positive semidefinite in R (V(0) = 0, V(x) > 0 for x E R, x =~ 0); V has continuous partial derivatives in R; 1k = d V / d t is negative semidefinite in R; (1:(0) = 0, F(x) < 0 for x ~ R, x 4= 0). An equilibrium point is locally asymptotically stable if and only if there exists a Liapunov function for it. Some necessary properties of Liapunov functions for Volterra systems are easily shown. Taking an arbitrary Liapunov function F(x), expansion in a Taylor series

101

up to terms of second order results in 1 V(x) : I.)0 + ~ l)ix i q- -~ ~ l),jxix j + ... i ij

The positive definiteness of V implies that v0 : V(0) = 0, v i = 0V/Oxi[ 0 : 0, and that the matrix W of components v i i = 021(./Oxi0xj[0 is itself positive definite. Similarly, the negative definiteness of V(x) implies that the matrix of components Xiv~jJ, k is negative definite. In matrix notation, these results can be summarized as: V(x) = x T W x -F 0(xi),3.

W is positive semidefinite; w TJ(E) + J(E)TW is negative semidefinite. The existence of a matrix W which fulfils the last of these conditions is, of course, equivalent to J(E) possessing only negative eigenvalues. For the nontrivial equilibrium E m, the following have been proposed as Liapunov functions: VI(X) = E Vli(Xi} I

with Vl~(xi) = x ~ - E m ln(xg/E m + 1), by Aiken and Lapidus (1973);

V2(x)= E ciV1i(xi} i

by Goh (1976);

V3(x) = Xc,x i

V4(x) =

f°

x. dx

by Gilpin (1974); and 1 ~5 (X) ~- ~" E 1)ijXiXj

in a special case, by MacArthur (1970). Tuljapurkar and Semura (1979a) introduced also the solution of a special partial differential equation as a Liapunov function. The same authors compared this type of Liapunov function with the types V~, V4 and V5 for two-population symmetric competition (Tuljapurkar and Semura, 1979b). An application of the above types of Liapunov function to trivial equilibria is possible if special restrictions for the vanishing components of these equilibria are taken into account. A comparative analysis of the Liapunov functions yields the following results.

102 V~ can be applied only for globally stable equilibria. Since its time derivative is 12t = Y,ijaux~x j exactly, the domain of attraction is either the whole positive quadrant (for negative definite matrix A) or empty. For conservative trophic systems (aij = -aj~), 12t = 0, i.e., E"' is neutrally stable and V~ is a constant of motion (see Kerner, 1971). No additive type of Liapunov function V = E i c , f ( x , ) , such as types V1-V4, is suitable for all locally stable equilibria. This may be shown as follows. The quadratic approximation of an additive Liapunov function is V3 with all cg I> 0 because of positive definiteness. The quadratic approximation of the time derivative is obtained as 12z--E~kciE ~ aij,x~x k. The matrix (c~E~'a~k } has to be negative definite. This is impossible for example for Allee-effect models, for which a~ > 0, which allow locally but not globally stable equilibria. V5 serves as a quadratic approximation for any Liapunov function. Even for two-population Allee-effect models, V5 captures all locally stable equilibria having parameters v l, = 1J211, v22 = I Jl2 J, and 1912 ~- /021 = [J12 J J21 I(I J~, I + 112=I)1/[21J = J2, I + ½(J,, + J==)=] where ,/,j = a u E r PERSISTENCE IN VOLTERRA MODELS Global stability or instability of the nontrivial equilibrium E"' is the most important feature of the Volterra model. Population systems having globally stable E m can be called persistent in the sense that no population is extinguished and no population grows without bound. Since most ecosystems studied (with the exclusion of some competitive systems) display at least medium-term persistence, nonpersistent Volterra models do not qualify as appropriate models for these systems. Thus the question arises of which Volterra models are persistent and which not. There are several approaches to answer this question: numerical study of model behaviour for given parameters; analysis by means of V~ (see preceding section) for given parameters; derivation of a set of conditions, as done by Goh (1976, 1977), - sufficient conditions for global stability of E " are: feasibility and local stability of E " and for all density feedbacks a , < 0: these conditions imply certain restrictions on the parameters of the community matrix A; derivation of conditions for interaction structures (i.e., configuration of zero and nonzero elements in the community matrix A) which allow pers i s t e n c e - n o t taking into account special parameter constellations. In the following some of the authors' results are given as obtained by the last approach, using a generalization of the competition-exclusion principle of Gause (1934), and the more general principle of Rescigno and Richardson (1973). The main result is a type of interaction-exclusion principle:

103

A Volterra system is not persistent if one of its subsystems contains more populations than the number with which it interacts.

Generally, the populations with which a subsystem interacts include prey, predators and competitors (outside and within the subsystem), and even populations of the subsystem having density feedback. The proof of the above result follows closely the approach of Rescigno and Richardson (1972). Let { N , / i E X} be a subset of state variables and the corresponding subsystem N i : Ni( ~i + ~ j ~ v a , j N j ) for i @ X. A necessary condition for persistence of the subsystem is card X ~< card Y, where Y : { j / B i ( i E X A a u vL 0)}. Let the subsystem be persistent. Then Ni(t) = 0 is excluded and it holds, with suitable reindexing, that iVi/Ni-~,i-k-ZaijNj

( i = 1 ..... m = card X)

J

Assume rank{a~j} < m. Consequently, the ruth line of the matrix {a~y} is a linear combination of lines 1 to m - 1, i.e., there exist linear coefficients /31..... fl,,-i such that fllal j + 1~2az j Jr- "" -[- ~m-- lain -

l,J

--

amy = 0

for allj E Y

Therefore with/3 m = - 1, it holds that Z ~ilV,//Ni = Z fli•i = constant i t

Integration and exponentialization yields

Excluding the improbable case that ~Bici = 0 , the right-hand side approaches either 0 or m for t ~ m. Therefore, at least one term of the left-hand side also approaches 0 or m. This is in contradiction to the persistence of the subsystem. Consequently, rank{ad} ~>m~ which implies card Y I> card X. Thus the principle is proved. The implications for trophic Volterra systems (having a u a j ~ < 0 ) are obvious. A t r o p h i c Volterra system is persistent only if any of its subsystems has more prey and predator populations than component populations (compare Steinmaller, 1980). It is interesting to apply this result to systems having k clear-cut trophic levels. Denoting by n~ the number of populations of level i, persistence implies the set of inequalities nk ~ n k _ I n~<<-ni_~+n~+~ n 1 ~n

2

for l < i < k

104

not

persistent

possible persistent

O

~D

with clear- cut trophic levels dP

0

YY without clear-cut trophic letters

e~r-e o o

trophic

interaction between t w o p o p u l a t i o n s

o populations

of a n o n - p e r s i s t e n t

subsystem

Fig. 5. Examples of nonpersistent and possibly persistent trophic Volterra systems.

and further,

E ni= ~ n, i odd

i even

Some solutions of this system have been given by Deakin (1971). Sinha (1978) has shown that for a system with two trophic levels, the condition //I =/72 holds also when time lags are included. Typical examples of nonpersistent and possibly persistent trophic Volterra systems are shown in Fig. 5. BEYOND THE VOLTERRA MODEL One of the disadvantages of Volterra models is that they are adequate only for ecosystems having one nontrivial equilibrium point. It has been

105

observed that for example forest-insect/insectivore systems show an interesting switching behaviour from one stable equilibrium to another when perturbed (Isaev et al., 1979). Analytical investigations of higher nonlinearities are generally possible only for small-dtmensional systems, and they are done by studying the phase representation (see for example, Bazykin, 1976). Simulation analysis becomes a necessity here as well as in the case of environmental forcing-terms with external dynamics. REFERENCES Aiken, R.C. and Lapidus, L., 1973. The stability of interacting populations. Int. J. Syst. Sci., 4: 691-695. Bautin, N.N., 1954. On periodical solutions of a special system of differential equations. Appl. Math. Mech., 18:128 (in Russian). Bautin, N.N. and Leontovich, G.A., 1976. Methods and Examples for Planar Qualitative Analysis of Dynamic Systems. Nauka, Moscow (in Russian). Bazykin, A.D., 1976. Structural and Dynamic Stability of Model Predator-Prey Systems. IIASA RM-76-8, International Institute for Applied Systems Analysis, Laxenburg, Austria. Coste, J., Preyraud, J., Coullet, P. and Chenciner, A., 1978. Quelques proprirtrs des systrmes de Volterra. J. Phys. (Paris), 39(C5): 43-47. Deakin, M.A.B., 1971. Restrictions on the applicability of Volterra's ecological equations. Bull. Math. Biophys., 33: 571-577. Dubois, D.M., 1979. State of the art of predator-prey systems modelling. In: S.E. Jorgensen (Editor), State of the Art in Ecosystem Modelling. Proc. Conf. Ecol. Modelling, Copenhagen, Denmark, 1978, pp. 163-217. Gause, G.F., 1934. The Struggle for Existence. Williams and Wilkins, Baltimore, MD. Gilpin, M.E., 1974. A Liapunov function for competition communities. J. Theor. Biol., 44: 35-48. Goel, N.S., Maitra, S.C. and Montroll, E.W., 1971. On the Volterra and other nonlinear models of interacting populations. Rev. Mod. Phys., 43: 231-276. Goh, B.S., 1976. Global stability in two species interaction. J. Math. Biol., 3: 313-318. Goh, B.S., 1977. Global stability in many-species systems. Am. Nat., 111: 135-143. Isaev, A.S., Chlebopros, R.G. and Nedorezov, L.V., 1979. Qualitative analysis of a phenomenological model for forest insect population dynamics. Preprint, Forest and Wood Institute, Krasnojarsk, U.S.S.R. (in Russian). Kerner, E.H., 1961. On the Volterra-Lotka principle. Bull. Math. Biophys., 23: 141-157. Kerner, E.H., 1971. Gibbs Ensemble, Biological Ensemble. Gordon and Breach, New York. MacArthur, R.H., 1970. Species packing and competitive equilibrium for many species. Theor. Pop. Biol., 1: 1-11. May, R.M., 1973. Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ. O'Neill, R.V. and Gardner, R.H., 1979. Sources of uncertainty in ecological models. In: B.P. Zeigler, M.S. Elzas, G.J. Klir and T.I. Oren (Editors), Methodology in Systems Modelling and Simulation. North Holland, Amsterdam, pp. 447-464. Patten, B.C., 1978. Necessary conditions for realism in ecosystem models. Paper presented at Symp. on Mathematical Modelling of Man-Environmental Interactions, Telavi, U.S.S.R. Rescigno, A. and Richardson, I.W., 1973. The deterministic theory of population dynamics. In: R. Rosen (Editor), Foundations of Mathematical Biology. Vol. III. Academic Press, New York, pp. 283-359.

106 Sinha, A.S.C., 1978. Two trophic levels model for population growth depending on past history. Int. J. Syst. Sci., 9: 1325-1330. Steinmiiller, K., 1980. Bilinear modelling in ecology. Proc. Symp. on Simulation of Systems in Biology and Medicine. Prague. Vol. 2, pp. 97-106. Svirechev, Ju.M. and Logofet, D.O., 1978. Stability of Biological Communities. Nauka, Moscow (in Russian). Tuljapurkar, S.D. and Semura. J.S., 1979a. Stochastic instability and Liapunov stability. J. Math. Biol., 8: 133-148. Tuljapurkar, S.D. and Semura, J.S., 1979b. Liapunov functions: geometry and stability. J. Math. Biol., 8: 25-32. Volterra, V., 1931. Lemons sur la Th~orie Mathematique de la Lutte pour la Vie. Gauthier Villars, Paris.