__M. MAKAN (mfn07)/ F. DEGHETTO (mfn04).__

Chaos theory attemts to explain the fact that complex and unpredictable results can and will occur in systems that are sensitive to their initial conditions.

A non linear demographic model was used to predict the
population dynamics of the flour beetle *Tribolium* under laboratory
conditions and to establish the experimental protocol that would reveal
chaotic behavior. With the adult mortality rate experimentally set high,
the dynamics of animal abundance changed from **equilibrium** to **quasi
periodic** cycles to **chaotic** as adult-stage recruitment rates
were experimentally manipulated, this transitions in dynamics corresponded
to these predicted by the mathematical model. Phase-space graphs of the
data together with the **deterministic** model attractors provide convincing
evidence of transitions to **chaos**.

The mathematical theory of non linear dynamics has led population biology into a new experimental and theoretical research. Explanation of observed fluctuations of population numbers now include dynamical regimes with a variety of asymptotic behaviors:

1. Stable equilibrium, in witch population numbers remain constant;

2. Periodic cycle, in witch population numbers oscillate among a finite number of values;

3. Quasi periodic cycle, witch are characterized by aperiodic fluctuations that are constrained to a stable attractor called an invariant loop;

4. Chaos, where population numbers change erratically and the pattern of variation is sensitive to small differences in initial conditions.

There is, however, a need for new experiments to confirm these hypothetical possibilities. A key theoretical prediction, witch is subject to experimental challenge, is that specific sequences of transition among qualitatively different dynamical regimes occur in response to changing biological parameters.

They modeled the relation of larval, pupae, and adult
*Tribolium* numbers at time t+1 to the numbers at time t by means
of a system of three stochastic difference equation :

In this model, L_{t} is the number of feeding
larvae (referred to as the L-stage) at time t;

P_{t} is the number of large larvae, non feeding
larvae, pupae , and callow adults (collectively the P-stage) at time t,

And A_{t} is the number of sexually mature adults
(A-stage animals) at time t.

The quantity b>0 is the number of larval recruits per adult per unit of time in the absence of cannibalism.

The fractions u_{l} and u_{a} are the
larval and adult rates of mortality in one time unit.

The exponential non linearities account for the cannibalism of eggs by both larvae and adults and the cannibalism of pupae by adults.

The fractions *exp(-c _{el}L_{t})*
and

The fraction *exp(-c _{pa}A_{t})*
is the survival probability of a pupa in the presence of A

The terms E_{1t },_{ }E_{2t }and
E_{3t }are random noise variables assumed to have a joint multivariate
normal distribution with a mean vector of zeros and a variance-covariance
matrix denoted by Z. the noise variable represent unpredictable departures
of the observations from the deterministic skeleton and are assumed to
be correlated with each other within a time unit but uncorrelated through
time. the deterministic skeleton of the model is identified by setting
Z=0, or, equivalently, by setting E_{1t },_{ }E_{2t
}and E_{3t} equal zero.

Experimentally, the parameter is estimated in deterministic skeleton to :

b=6.598, c_{el}=0.01209 , c_{ea}=0.01155,
u_{l}=0.2055 and u_{a}=0.96.

With the parameter values indicate before, a sequence
of transitions from a stable equilibrium to quasiperiodic and periodic
cycles to chaos is : c_{pa}= 0.0, 0.05, 0.1, 0.2, 0.45, 0.75, 1.0
.

The Cpa=0.0 treatment is a region of stable equilibrium, the Cpa=0.05 and Cpa=1.0 treatment are in regions of stable equilibrium, the Liapunov exponents are negative.

For the Cpa=0.1 treatment the Liapunov exponent is 0, witch is consistent with a quasi periodic attractor.

For Cpa=0.0, the model predict an oscillatory approach to equilibrium, with approximately equal numbers of insects in all tree life stage.

For Cpa=0.1, a distinctive 3-cycle is predicted, with a repeating high-low-low pattern.

Thus, a positive Liapunov exponent is a signature of chaos. in this model there is different region with multiple attractors, with a stable 3-cycle coexisting with chaos or stable cycles. In this regions the asymptotic dynamic depends on the initial condition.

Contrary to the idea that chaotic dynamics may lead to extenction, the bifurcation diagram reveals that extinction is unlikely for these populations even in regions of chaotic dynamics.

under certain conditions, the motion of a particle described by certain systems will neither converge to a steady state nor diverge to infinity, but will stay in a bounded but chaotically defined region.

The experimental confirmation of nonlinear phenomena in the dynamics of the laboratory beetle lends credence to the hypothesis that fluctuations in nature populations might often be complex, low-dimensional dynamics produced by nonlinear feedback. In this example, complex dynamics were obtained by "harvesting" beetles to manipulate rates of adult mortality and recruitment.

For applied ecology, the experiment suggests adopting a cautious approach to the management or control of natural populations. In a poorly understood dynamical population system, human intervention -such as changing a death rate or a recruitment rate- could lead to unexpected and undesired results.