Practical use of adaptation principles in ecological systems modelling

PRACTICAL USE OF ADAPTATION PRINCIPLES IN ECOLOGICAL SYSTEMS MODELLING

Peter Racsko

1. Introduction The term "adaptation" is widely used in ecological, biological, economical systems research as a qualitative characteristic of complex sy&ems~ Generally speaking adaptation expresses the system's ability to reduce thq amplitude of responses to the changes of the embedding system i.e. er~vir~timental conditions. While the adaptation of biological objects is quite explicit and understood as a tool for survival of individuals and populations, for ecological systems with trophical structure, systems with external control - e.g. human impact - adaptation is treated as a type of behaviour oriented to preserve stability. The term stability is used in connection with the steady latatesof the system and stands for the system's ability to resist to not very significant perturbations. We shall demonstrate a pair of examples, one of them illustrates the practical use of a hypothesized adaptation principle in a tree growth simulation, the other one proves the fact that a forced adaptation can destabilize a highly developed ecological production system.

e. A tree growth simulation model The whole model description was published in Cll, only a simplified version will be demonstrated below. Let xl, xa, xa be the root, bole and leave biomass of a tree respectively. Let y be the new organic matter produced by the photosynthesis during the time period T. The system's evolution can be described by the follouimg system of difference equations: Xi(t

+ ‘cl = xi(t)

+ ei(t) y(t, X)

i = 1, 2, 3 with the initial conditions xi(O) = x1, (i=1,2,3), where 1)

Computing Centre of the EBtvos Lorand University 1117. Budapest, Bogddnfy u. IO/b.

224

eit

- the distribution rate of the new assimilates between the parts of 3 the tree; X ei(t) = 1; ei(t) 2 0. i=l

X(t)

- vector of the environmental

conditions.

y(t, T) - is defined as follows: y(t,h)

=

y(x,(t),

x2(t),

x3(t),

(2)

X(t))

As one can see system (1) is not "closed", i.e. the ei(t) functions can not be obtained from the system itself. In order to "close" the system we postulate an adaptation hypothesis, that the system distributes assimilates

in a way that maximizes

the new

the overall biomass production

next time period supposing the environmental

in the

conditions do not change.

The functions e,(t), e=(t), eJ(t) are defined from the expression: y,(t+r) =

max y(x,(t)+er(t)y(t,~)T, Iel(t),ea(t),e3(t)

. . . , xa(t)+e3(t)y(t,X)r,

.. . .

(3)

X1

3 C ei(t) = 1 . i=l Now the system (1) with the adaptation principle complex but conceptually

(3) can be solved. A more

the same model was identified and investigated

for pine forests of Carelia. The computer simulation results occured to be sufficiently adequate. The adaptation principle provides a numerical technique to investigate the limits of survival of a tree as a part of the forest biogeocenosis.

The simulated system is very stable and tolerates

relatively large changes of the exogenous parameters e.g. wheather conditions, nutrient and water quantity etc.

3. Ecosystem stability with external control The second example demonstrates

that an externally

supported equilibrium

state of an ecological system is unstable if the system is relatively well-developed

and the output-control

strategy is fixed.

In this case the system can not tolerate even small perturbations,

i.e.

it will never be adapted to the artificial equilibrium. The ecological system is a two-level trophical chain (resource and consumer) with controlled resource input flow and biomass output.

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The system's model: dR x = Q(R)

-

mR - NV(R) (p(G) (4)

dN - = aNV(E) up dt

- S(N)

where R - resource quantity, N - consumer

(biomass or production) quantity,

Q(R) - resource input function, S(N) - production output function, m - natural loss rate of resources, V(R) - trophical function, a - conversion parameter and (p(v) - parameter expressing the impact of the external factors i;. In real production systems the following conditions hold: i) Q(R); S(N) and V(R) are sufficiently smooth functions inlR1 (p(i) is sufficiently smooth inIR" ii) V(0) = 0; V(R) I M for every R 2 0 g

2 0 for every R 2 0

iii) 0 I (p(L) 5 1 Assume the production output quantity has been selected and fixed, S(N) = C. The necessary and sufficient conditions for the stability of a steady state (R*, N*) are: dQ

zI

-m>O

dQ =

- m +

(51

R*

I

VCR*) cp(F) - 5 * &I

* %I

RC

< 0 R*

As far as Q(R) is an external control (5) can always be satisfied however is "large" i.e. when

(6) holds only for "small" VCR*) values, when g the system is in its early period of evolution.

R"

During this period proper selection of Q(R) provides a stable equilibrium. For "large" V(R) values there are no control strategies providing stability. The changes of the environmental

factor (p(G) might shift the

separating line between the domain of possible stable steady states and the part of the phase plane where stable steady states can not exist. As we have seen from (6) in this type of forced ecological systems the 1 dV When this value W';iii. is small there is no way to maintain a stable equilibrium, i.e. the

stability strongly depends on the proportion

system loses the ability to adapt to small perturbations.

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Consequently,

computer simulations based on this type of models have

to be performed very carefully.

Budapest 30. 01. 1985.

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