FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES

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1 FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES The goal of these notes is to illustrate an application of large powers of matrices Our primary tools are the eigenvalues and eigenvectors of the matrix We illustrate this with two familiar examples Fibonacci s numbers We are all familiar with Fibonacci s sequence,,,, 3,, 8, 3,, What if we wanted to compute quickly (this is the keyword!) the th Fibonacci s number? Here is how matrices can help us Write f, f, f, f 3, f 4 3, f, f 6 8, f 7 3, In other words, Fibonacci s numbers are given by the recursive relation f n+ f n+ + f n for n, with f and f Notice that That is, we can write the previous expressions as f f f3 f f f f f 3 f4 f 3 From this it also follows that 3 f3 f f4 f f f f 3 f fn+ In general we see that if we set u n we have the recursive relation f n () fn+ fn+ u n+ Au f n+ f n n u A f3 We can also solve () explicitly and produce the solution in terms of the powers of the matrix A That is () u n A n u u Notice that () gives us the transition (relation) between consecutive Fibonacci s numbers More specifically, the equation u n+ Au n for n encodes the relation f n+ f n+ + f n (the second relation encoded is the tautology f n+ f n+ ) Notice that () gives us a way to calculate u n (that is f n and f n+ ) from u (that is f and f ) by means of the equation u n A n u Let us compute the eigenvalues and eigenvectors of the matrix A introduced above Despite the fact that A is rather simple, the eigenvalues and the eigenvectors of A are not nice at all! The characteristic polynomial of A is det λ Hence we obtain the following eigenvectors λ det λ λ λ λ, ± f ( λ)( λ) λ λ Date: March 4, 3

2 FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES λ + : In order to find (one of) the eigenvector(s) v associated to λ we need to solve the following system of equations Av λ v (A λ I )v That is we need to find the solutions of + + } {{ } A λ I a b v Since we are subtracting one of the two values that make the matrix A singular, we have that the system of two linear equations in a and b reduces to the single equation a + b If we set b then a + Hence we get the eigen pair λ + v + λ : In order to find (one of) the eigenvector(s) v associated to λ we need to solve the following system of equations Av λ v (A λ I )v That is we need to find the solutions of } {{ } A λ I c d v Since we are subtracting the other of the two values that make the matrix A singular, we have that the system of two linear equations in a and b reduces to the single equation c d If we set d then c Hence we get the eigen pair λ v Let us now rewrite the vector u as a linear combination of the eigenvectors v and v That is we are seeking values c and c such that c v + c v u + + c + c c + c c + c

3 FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES 3 + c c This system of linear equations leads to the solutions c / and c / That is u v v Hence the relation u n A n u translates into the following ( u n A n v ) v u n A n v A n v u n λ n v λ n v f n+ f n ( ) + n + ( ) n (In other words we have u n An u A n (c v + c v ) c A n v + c A n v c λ n v + c λ n v ) The above matrix equation translates into the following two (consistent) expressions f n+ ( ) + n+ ( ) n+ and f n ( ) + n ( ) n Observe that the eigenvalue λ ( + )/ 68 (the largest, or dominant, eigenvalue) and the eigenvalue λ ( )/ 68 Hence λ n (in an oscillatory fashion) as n Thus we conclude that (for n sufficiently large) f n closest integer to ( ) + n The eigenvalue λ ( + )/ is also called the golden ratio Let us check the above result in the chart below n ( ) + n f n

4 4 FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES A population model If we analyze our description of the Fibonacci s numbers, we realize that fn+ we had a matrix A that was giving us a transition between a set/state u n to the next f n fn+ set/state u n+ Thus, what we did for the Fibonacci s numbers can be applied to f n+ describe the dynamics of a population that have a finite number (not necessarily two) of stages in life Suppose that we consider an hypothetical animal that has two life stages: juvenile adult Suppose that we count the number of members of this population on a weekly basis Say J t number of juveniles at week t A t number of adults at week t The relation between the population during two consecutive weeks can reasonably be described as follows (3) J t+ J t mj t gj t + fa t, where the term mj t accounts for the fraction of juveniles that dies, the term gj t accounts for the fraction of juveniles that becomes adult, and the term +fa t accounts for the newborns; (4) A t+ A t µa t + gj t, where the term µa t accounts for the fraction of adults that dies and the term +gj t accounts for the fraction of juveniles that becomes adult Observe that m, g, f, µ are numbers that denote weekly rates, which we assume to be constant for each period We can express express the relations described above in matrix form as + m g f () A t+ g µ A t A state t state t+ If we define the vector u t for any integer t then we can rewrite the above expression A t in the following recursive way J u t+ Au t with u ; A in an explicit form we have u t A t u with u J A Since we are interested in the dynamics of this population we have another example of large powers of a matrix Example: We illustrate the above model with a numerical example Suppose that g m, f, and µ 9 Hence () becomes u t+ u t We can easily check that the eigen pairs are 99 λ v λ 9 v As we discussed in the case of the Fibonacci s numbers, the general solution to our problem is u t A A t u A t (c v + c v ) c A t v + c A t v c λ t t v + c λ t v,

5 FIBONACCI S NUMBERS, A POPULATION MODEL, AND POWERS OF MATRICES where c and c are the values that allow us to rewrite the vector u as a linear combination of the eigenvectors v and v As it happened in the case of the Fibonacci s numbers, one of the eigenvalues is dominant Namely, λ > 9 λ Thus we can rewrite our solution as ( ( ) t ) u t λ A t λ c v + c v t λ as t λt c v It makes sense to call the dominant eigenvalue λ the growth rate (λ growth rate %) and the corresponding eigenvector v the stable age structure Also observe that the second term in the general solution leads to an oscillating (decaying) behavior caused by the factor ( 9) t As we observed earlier in the long run we have u t c A () t 99 t This implies that the ratio J t c () t 99 A t c () t 99 is constant This means that in the long run the population will consist of 66% of juveniles and 344% of adults In other words there will be about 9 juveniles for every adult Remark The above population model is an example of a Leslie matrix You can read more about Leslie matrices (even for populations with more than two life stages!) on pages and of our textbook Calculus for Biology and Medicine by Claudia Neuhauser The example is taken from the book Mathematical Methods in Biology by JD Logan and W Wolensky (pages 3-) References E Batschelet, Introduction to Mathematics for Life Scientists L Edelstein-Keshet, Mathematical Models in Biology 3 DE Goldberg and JH Vandermeer, Population Ecology 4 JD Logan and W Wolesensky, Mathematical Methods in Biology JD Murray, Mathematical Biology 6 C Neuhauser, Calculus for Biology and Medicine 7 C H Taubes, Modeling Differential Equations in Biology If x represents the percentage of juvenile then x represents the percentage of adults Hence the equation (ratio) x/( x) 99 gives the solution x 99/99 66 and x 344