Modeling the population dynamics and community impacts of Ambystoma tigrinum: A case study of phenotype plasticity

Abstract

Phenotypic plasticity is the ability of an organism to change its phenotype in response to changes in the environment. General mathematical descriptions of the phenomenon rely on an abstract measure of "viability" that, in this study, is instantiated in the case of the Tiger Salamander, Ambystoma tigrinum. This organism has a point in its development when, upon maturing, it may take two very different forms. One is a terrestrial salamander (metamorph)that visits ponds to reproduce and eat, while the other is an aquatic form (paedomorph) that remains in the pond to breed and which consumes a variety of prey including its own offspring. A seven dimensional nonlinear system of ordinary differential equations is developed, incorporating small (Z) and large (B) invertebrates, Ambystoma young of the year (Y), juveniles (J), terrestrial metamorphs (A) and aquatic paedomorphs (P). One parameter in the model controls the proportion of juveniles maturing into A versus P. Solutions are shown to remain non-negative. Every effort was made to justify parameters biologically through studies reported in the literature. A sensitivity analysis and equilibrium analysis of model parameters demonstrate that morphological choice is critical to the overall composition of the Ambystoma population. Various population viability measures were used to select optimal percentages of juveniles maturing into metamorphs, with optimal choices differing considerably depending on the viability measure. The model suggests that the criteria for viability for this organism vary, both from location to location and also in time. Thus, optimal responses change with spatiotemporal variation, which is consistent with other phenotypically plastic systems. Two competing hypotheses for the conditions under which metamorphosis occurs are examined in light of the model and data from an Ambystoma tigrinum population at Mexican Cut, Colorado. The model clearly supports one of these over the other for this data set. There appears to be a mathematical basis to the general tenet of spatiotemporal variation being important for the maintenance of polyphenisms, and our results suggest that such variation may have cascading effects on population, community, and perhaps ecosystem dynamics because it drives the production of a keystone, cannibalistic predator.

Fuzzy Differential Equations as a Tool for Teaching Uncertainty in Engineering and Science

“…This manuscript reports fuzzy differential equations as an option to teach uncertainty in engineering and sciences. The case of study reported in this paper is to teach the Cauchy problem, in particular, in the still open problem of population dynamics [20], specifically about the classic Malthusian demography model, to induce the need of studying about uncertainty in the fuzzy context.…”