Population Dynamics Models

Lotka-Volterra Model

The Lotka-Volterra model is a very simple and traditional model of population dynamics between preys and predators. In its simplest form, describing one species each of prey and predator, the model is characterized by two coupled first order Ordinary Differential Equations:

$$\begin{eqnarray*} \frac{d x(t)}{d t} & = & x(t) \left( A - B y(t) \right) \\ \frac{d y(t)}{d t} & = & - y(t) \left( C - D x(t) \right) \end{eqnarray*}$$

The populations of the two species are denoted $x(t)$ and $y(t)$ respectively. The coefficients $A$, $B$, $C$ and $D$ are constants representing rates. They are fitted by using data from observations of specific species.

This model is called the Lotka-Volterra Two Species Model, the Lokta-Volterra Predator Prey Model, or the Resource Consumer Model in the world of business. Details of the model can be found in numerous web sites, in particular those by Wilkepedia( Lotka-Volterra-equation ) and Wolfram Lotka_Volterra equations.

It is simple to adapt the mass-on-spring ODE RK4 solver code to solve the Lotka-Voltera model. The code LV.c does precisely this for the parameters indicated in the web site by Wolfram.

Fixed Points

The Fixed Points of a system of ODEs are those points in phase space where the time-derivatives are all zero simultaneously. Evidently, these points are singular in that they do not evolve in time into other points in phase space, or, more precisely, they evolve into themselves.

It can be seen that these Fixed Points organize the flow resulting from the ODEs. This happens through the fact that these points organize the linear behavior of the local flow around them. The mathematically inclined reader can read a general description of Fixed Points and a derivation of their properties in Nonlinear Dynamics - A Two Way Trip from Physics to Math by H.G. Solari, M.A Natiello G.B. Mindlin

The Lotka-Voltera model admits two fixed points: $(x,y)=(0,0)$ and $(x,y)=(C/D,A/B)$. The code LV.c can be used to gain intuition on how these points organize the flow in this model.

  

  

  

A Non-Linear Model

Another model used in population dynamics, as well as to describe competition between firms in the business world, is described by the following Ordinary Differential equations

$$\begin{eqnarray*} \frac{ d u1(t) }{ d t } & = &u1(t) ( 1 - u2(t) u2(t) ) - u2(t) \\ \frac{ d u2(t) }{ d t } & = & u1(t) - u1(t) u2(t) \end{eqnarray*}$$

The functions to solve for are $u1(t)$ and $u2(t)$ given their initial values $u1(0)$ and $u2(0)$ at time $t=0$. This model is a modified version of the Lotka-Volterra Two Species Model.

It is simple to adapt the mass-on-spring ODE RK4 solver code to solve this non-linear model. The code LV2.c does precisely this. There are no explicit parameters to adjust in this model.

Fixed Points and Limit Cycles

The Fixed Points influence the behavior of the ODEs flow in phase space. This is certainly the case for this model. But trajectories launched from different points behave very differently than in the simple Lotka-Volterra model. Some initial conditions produce trajectories that converge toward a periodic trajectory independently of the initial conditions. This periodic orbit is called a Limit Cycle. The Limit Cycle is an Attractor. The trajectories that reach it do so after some transient time. For instance, launch trajectories from $(u1(0),u2(0)) = (0.1,0.1)$ and $(u1(0),u2(0)) = (0.0,-0.5)$

Some initial conditions lead to trajectories that latch onto the Limit Cycle. Others lead to trajectories that escape to infinity. Therefore, this leads to the notion of a Basin of Attraction. This is the region (set of points) in phase space that lead to trajectories that converge toward the Limit Cycle. The Basin of Attraction can be found numerically by trials and errors or by systematic scanning.