Chaotic Dynamics in an Insect Population.

M. MAKAN (mfn07)/ F. DEGHETTO (mfn04).

1. Introduction.

2. Mathematical model.

3. Results.

4. Commentaries.

5. Conclusion.

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1. Introduction.

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.

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2. Mathematical model.

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, Lt is the number of feeding larvae (referred to as the L-stage) at time t;

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

And At 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 ul and ua 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(-celLt) and exp(-ceaAt)are the probabilities that an egg is not eaten in the presence of Lt larvae and At adults in one time unit.

The fraction exp(-cpaAt) is the survival probability of a pupa in the presence of At adults in one time unit.

The terms E1t , E2t and E3t 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 E1t , E2t and E3t equal zero.

Experimentally, the parameter is estimated in deterministic skeleton to :

b=6.598, cel=0.01209 , cea=0.01155, ul=0.2055 and ua=0.96.

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3. Results:

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

 

 

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4. Commentaries:

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.

 

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5. Conclusion.

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.

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