MI-Sim A MATLAB Package for the Numerical Analysis of Ecological Motifs

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• Basin of attraction: A set of points (initial conditions) in the phase space from which a dynamical system spontaneously moves to a particular attractor.

• Bifurcation: A sudden qualitative change in the behaviour of a dynamical system given a small change in the system parameter values. A critical bifurcation can be indicative of a significant positive or negative transition between steady-states.

• Dilution rate: Inflow of substrate, nutrient or medium per volume of reactor or culture.

• Eigenvalue: The value of a parameter for which a differential equation has a non-zero solution for given conditions. Used to determine the stability of fixed-points from the Jacobian of the ODEs.

• Fixed-point: The fixed-point of a function is such that it maps to itself in the functional domain, e.g. for a function $f(x)$, $p$ is a fixed-point if $f(p) = p$. In stability theory for dynamical systems, fixed-points can represent equilibrium positions in a autonomous system of differential equations (i.e., $f(x) = 0$). Stability may be determined by examining the response to small perturbations from the fixed-points.

• Gibbs Free Energy: The energy associated with a chemical reaction that can be used to do work. A negative Gibbs Free Energy ($\Delta$G) indicates a favourable reaction can (spontaneously) take place depending on temperature.

• Jacobian Matrix: A matrix of first-order partial differential equations of a set of a vector function (i.e., in MI-SIM, the vector of model ODEs). Eigenvalues of the Jacobian determine the behaviour of trajectories close to the system equilibria (fixed-points). These trajectories can be thought of as perturbations in the parameter space containing the fixed-points. The trajectories, evaluated as a function of time, will determine if the equilibria are stable or unstable.

• Linear Stability Analysis: Linearisation of a non-linear function close to a given point (e.g., fixed- or equlibrium-point). This can be done by calculating the eigenvalues of the Jacobian matrix for all $f(x)_n = 0$.

• Motif: The module or simple network describing the interaction between biotic and abiotic components and reactions in the modelled system.

• Phase portrait: A graphical representation of the dynamical trajectories or bifurcations occurring in the phase plane generated by the solutions to the system of ODEs.

• Routh-Hurwitz Stability Criterion: A mathematical method to determine whether all the roots of a polynomial lie in the left-half plane (i.e., if all the eigenvalues of the dynamical system are negative, then the fixed-point is stable).

• Steady-state: A process or system is at steady-state if the variables that describe its behaviour are constant with time. For example, in a biological system, this may occur in a chemostat when one organism is washed-out from the system and the other reaches equilibrium between removal and growth.