Two-parameter bifurcations in LPA model.

The structured population LPA model is studied. The model describes flour beetle (Tribolium) population dynamics of four stage populations: eggs, larvae, pupae and adults with cannibalism between these stages. We concentrate on the case of non-zero cannibalistic rates of adults on eggs and adults on pupae and no cannibalism of larvae on eggs, but the results can be numerically continued to non-zero cannibalism of larvae on eggs. In this article two-parameter bifurcations in LPA model are analysed. Various stable and unstable invariant sets are found, different types of hysteresis are presented and abrupt changes in dynamics are simulated to explain the complicated way the system behaves near two-parameter bifurcation manifolds. The connections between strong 1:2 resonance and Chenciner bifurcations are presented as well as their very significant consequences to the dynamics of the Tribolium population. The hysteresis phenomena described is a generic phenomenon nearby the Chenciner bifurcation or the cusp bifurcation of the loop.

Homeostatic model of human thermoregulation with bi-stability.

All homoiothermic organisms are capable of maintaining a stable body temperature using various negative feedback mechanisms. However, current models cannot satisfactorily describe the thermal adaptation of homoiothermic living systems in a physiologically meaningful way. Previously, we introduced stress entropic load, a novel variable designed to quantify adaptation costs, i.e. the stress of the organism, using a thermodynamic approach. In this study, we use stress entropic load as a starting point for the construction of a novel dynamical model of human thermoregulation. This model exhibits bi-stable mechanisms, a physiologically plausible features which has thus far not been demonstrated using a mathematical model. This finding allows us to predict critical points at which a living system, in this case a human body, may proceed towards two stabilities, only one of which is compatible with being alive. In the future, this may allow us to quantify not only the direction but rather the extent of therapeutic intervention in critical care patients.

Bifurcation manifolds in predator-prey models computed by Gröbner basis method.

Many natural processes studied in population biology, systems biology, biochemistry, chemistry or physics are modeled by dynamical systems with polynomial or rational right-hand sides in state and parameter variables. The problem of finding bifurcation manifolds of such discrete or continuous dynamical systems leads to a problem of finding solutions to a system of non-linear algebraic equations. This approach often fails since it is not possible to express equilibria explicitly. Here we describe an algebraic procedure based on the Gröbner basis computation that finds bifurcation manifolds without computing equilibria. Our method provides formulas for bifurcation manifolds in commonly studied cases in applied research - for the fold, transcritical, cusp, Hopf and Bogdanov-Takens bifurcations. The method returns bifurcation manifolds as implicitly defined functions or parametric functions in full parameter space. The approach can be implemented in any computer algebra system; therefore it can be used in applied research as a supporting autonomous computation even by non-experts in bifurcation theory. This paper demonstrates our new approach on the recently published Rosenzweig-MacArthur predator-prey model generalizations in order to highlight the simplicity of our method compared to the published analysis.