Emerging ecological patterns from optimal investment strategies in a randomly fluctuating environment

Taylor’s Law (TL) states that the variance of a random variable scales like a power law of its mean. Also a generalized version can be formulated extending the power-law scaling to the moments. In ecology they apply to populations and they are very corroborated empirical patterns shared among different ecosystems. The empirical evidences coming from the measurements provide TL exponent to be within 1 and 2, and more often to cluster around 2, while for the exponent relating the k-th to the j-th moment only values close to k/j are observed. In contrast theoretical models predict an unlimited range of values for both of these exponents. In this thesis, adopting the framework of multiplicative growth models in a Markovian environment, I investigate the possibility of the introduction of evolutionary strategies into the population dynamics to be the mechanisms that makes the exponents to fall into finite ranges. I implement three different strategies the individuals can follow and for the last two I performed three investment optimizations with different ecological goals. In all the analyzed cases I find TL exponent can assume any real value due to the existence of regions of the model parameters in which the exponent can diverge. Furthermore, under natural hypothesis on the dynamics of the environment and the range of parameters, the shapes of these regions do not depend on the strategy adopted and nor on the optimization objective. Thus the introduction of strategies does not affect the range of TL exponent in the model. In our theoretical framework rare events are shaping the value of the TL exponent, suggesting, as hinted by previous works, that empirical values may be a statistical artifact following from under sampling.