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Unformatted text preview: 2.5: AUTONOMOUS EQUATIONS AND POPULATION DYNAMICS KIAM HEONG KWA We will define the (asymptotic) stability of an equilibrium solution of an autonomous first order equation. We will also state two stability tests for testing the (asymptotic) stability of such an equilibrium solu- tion. Some applications of the notion of stability are illustrated in the homework problems. Consider the first-order autonomous equation (1) dy dt = f ( y ) ,y (0) = y , where the rate function f ( y ) is continuously differentiable, i.e., both f ( y ) and df dy are also continuous. From section 2.4, we know that there is always a locally unique solution ( t ) such that (0) = y and ( t ) = f ( ( t )) for all t in a sufficiently small neighborhood of t . In fact, if f ( y ) 6 = 0, then we can calculate the solution by separating the variables, i.e., Z dy f ( y ) = Z dt = t + c, the result of which is equivalent to (2) ( t ) = y + Z t f ( ( s )) ds. However, note that (2) includes also all possible equilibrium solutions, i.e., constant solutions ( t ) = y such that f ( ( t )) = f ( y ) = 0 for all real t . Rather than finding the solution of (1) explicitly, we are more inter- ested in the qualitative properties of the solution in many applications. For instance, let the solution ( t ) of (1) be the population of a species at time t . Then we may wish to answer the following questions. Date : January 2, 2011. 1 2 KIAM HEONG KWA (a) Does there exist an initial condition y at which the species exists in a steady state? That is to say, is there a number y such that ( t ) = y is a constant solution? Such a solution, as we have seen, is called an equilibrium solution of (1). It is also called a critical point of f ( y ). It can be found by locating the zeros of the algebraic equation f ( y ) = 0....
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