Linearizing nonlinear systems of equations. Example: predatory and prey populations

Transcription

1 Linearizing nonlinear systems o equations Aaron Fenyes (aenyes@mathtorontoedu) November 22, 28 Examle: redatory and rey oulations Put some yeast in a jar ull o nutrients Let to themselves, the yeast reroduce at a rate roortional to their oulation Y Their oulation growth is described by the equation Y Y Take another jar ull o Paramecia, which are microscoic redators Without rey to eat, the Paramecia are dying at a rate roortional to their oulation Their oulation P shrinks according to the equation P P Now, let the Paramecia into the jar o yeast, so they can eat the yeast The rate at which they eat is roortional to Y P The more Paramecia there are, the aster they can catch the yeast The more yeast there are, the easier they are to catch The oulations o yeast and aramecia are now linked by the system o equations Y Linearization around (, ) Y Y P P P + Y P The system above is nonlinear: Y and P aren t just weighted sums o Y and P When we zoomed in on the hase ortrait, however, we saw that near (, ) it looks like the hase ortrait o a linear system a saddle To understand what s going on, suose the oulation vector (Y, P ) is very close to the critical oint (, ) To be ancy, we can write [ Y P +, the assumtion that

2 Rewriting our system o equations in terms o the new variables y and, we get which simliies to ( + y) ( + y) ( + y)( + ) ( + ) ( + ) + ( + y)( + ), y y y + y Since y and are very small, their roduct y is much smaller than either o them Hence, the aroximation y y is very accurate In summary, near the critical oint (, ), our original nonlinear system o equations acts a lot like the linear system [ The hase ortrait o this linearized system is a saddle, exlaining why the hase ortrait o the original system looks like a saddle near (, ) The general solution o the linearized system is [ Ae t + Be t [ or constants A and B No matter what the initial conditions are, the yeast oulation grows exonentially, while the Paramecium oulation shrinks exonentially The Paramecia can t catch the yeast ast enough to sustain their oulation, or to slow down the yeast s exonential growth Linearization around (, ) Our nonlinear system has another critical oint at (, ) When we zoom in on the hase ortrait near that critical oint, we see what looks like the hase ortrait o a linear system a center [Show icture in notebook Once again, we can understand what s going on using linearization Suose [ Y P [ +, 2

3 Rewriting our system o equations in terms o the new variables y and, we get which simliies to ( + y) ( + y) ( + y)( + ) ( + ) ( + ) + ( + y)( + ), y y y + y Like beore, we observe that y is very small comared to y and, so the aroximation y is very accurate In summary, near the critical oint (, ), our original nonlinear system o equations acts a lot like the linear system [ [ [ y y The hase ortrait o this linearized system is a center, exlaining why the hase ortrait o the original system looks like a center near (, ) Examle: eidemic Our redator-rey oulation model has just a ew searate critical oints Most nonlinear systems look like that Sometimes, though, you might see weirder-looking hase ortraits more critical oints A ew lectures ago, or examle, we looked at a model or the sread o an inection, using the variables S, F, R or the oulations o suscetible, inected, and recovered eole S 3SF F R 2F y 3SF 2F Since R doesn t inluence S, and F, we don t have to kee track o it to understand how S and F are changing [Show hase ortrait in notebook S 3SF F 3SF 2F For this equation, every single oint on the line F is critical We can get a linear system by zooming in on any o these oints 3

4 Linearization around (, ) Earlier, or examle, we zoomed in on the critical oint (, ) Suose [ [ [ S s +, F Substituting, [ s [ ( + s) 3( + s)( + ) ( + ) 3( + s)( + ) 2( + ) Simliying, s 3 3s + 3s, so we have the aroximation s 3 The linearized system is [ s [ 3 [ s [Show zoomed-in hase ortrait The general solution is [ [ [ s A + Be t 3 or constants A and B Note how the number o inected eole grows exonentially Examle: endulum I mentioned at the beginning o the course that the angle o a endulum is described the second-order nonlinear equation X sin X 4

5 We ve been looking at this system or very small values o X to be ancy, X + x or x We used the aroximation sin x x to get the linear equation x x Let s look at this model in more detail Introduce the new variable V X to get the nonlinear system X V V sin X The critical oints are (nπ, ) or integers n [Show hase ortrait, i ossible It s not in the notebook yet, unortunately Linearization around (, ) Let s zoom in on the critical oint (, ), suosing [ [ [ X x +, V v Substituting, we get [ x v X + v V sin( + x) Using the linear aroximation sin( + x) sin + x sin x cos x or the sine unction, we get the aroximate linear system x v v x Linearization around (π, ) [Reeat 5