Ordinary differential equations: Slope field and solution curves
Behaviour of solutions
The slope field of a differential equation and solution curves ploteed herein often give a good first impression of the behaviour of solutions of the differential equation. We illustrate this by means of logistic growth model. In the next paragraph wediscuss the behaviour of solutions.
The growth model with an exponential increase in quantity \(y\) does not apply if there is a natural limit. The growth of a population may, for example, be limited by the amount of available food. A possibly better description is then given by a model in which the growth factor decreases with the size of \(y\). The simplest model is built on the assumption of a linear decrease. This is the so-called logistic growth model, proposed by the demographer Pierre François-Verhulst in order to counteract the growth of populations at high densities: \[\frac{\dd y}{\dd t}=r\cdot y\cdot\left(1-\frac{y}{a}\right)\] where \(r\) and \(a\) are nonzero constants. The constant \(a\) is called the carrying capacity, and \(r\) the intrinsic growth coefficient. The solution of this ODE with \(a=1\), \(r=1\) and initial value \(y(0)=\tfrac{1}{2}\) is the logistic function or so-called sigmoid function. This function, whose function rule is \(f(t)=\dfrac{1}{1+e^{-t}}\), and functions with a similar S-shaped function graph are much used in biological sciences; for example, in studies of bacterial growth.
The ODE can also be solved exactly: for example, the drawn solution of the ODE with initial value \(y(0)={{3}\over{2}}\) is in fact the graph of \(\displaystyle y(t)=\frac{3}{1+e^{-2 t}}.\)
Experiment with the solution curve and explore how the lineal elements in the slope field are directed in order to discover the behaviour of the solution curve.
If \(y(0)=3\), then the solution curve is a horizontal line because the corresponding function is constant. It is called a steady state, steady-state, or an equilibrium state.
If \(0\lt y(0)\lt 3\), then the solution curve is the graph of an ascending function which is going to run more and more flat in both directions when it gets closer to the horizontal lines \(y=0\) and \(y=3\). One says that the equilibrium \(y=3\) is an attracting equilibrium because every solution curve that is close enough to the steady state approaches in the course of time this equilibrium.
A negative starting value means a descending solution curve which is becoming farther away from the horizontal axis. The equilibrium \(y=0\) is a repelling equilibrium because every solution curve this only slightly deviates from it will over time move qaway from this equilibirum.
