From: "Richard A. Hinrichsen" Subject: Population Waves Date: Wed, 29 Sep 1999 15:54:46 +0000 Newsgroups: sci.math Keywords: population dynamics, Leslie matrices I have a question about population waves. Leslie matrices are being used by the National Marine Fisheries Service to model the dynamics of salmon populations that are nearly extinct. The goal is to find strategies that increase the Perron eigenvalues. It occurred to me that population waves will also be important in determining whether a population will persist. The difference equation governing the dynamics of the age-structured population is: v(t+1) = L*v(t), where L is a Leslie matrix, and v(t) is the age-specific population vector for year t (the intial age-specific vector is v(0)). As usual, the first row of L contains the age-specific fecundities, and the diagonal below the main diagonal contains the age-specific survival probabilities (See Leslie 1945 for further information). The entries of the nxn matrix L are nonnegative as are the entries of the nx1 vectors v(0),v(1),... . I wonder what form of the initial age population vector gives rise to the greatest oscillations? This could be used to get an upper bound on the importance of fluctuations to extinction risk. The matrices we use are primitive, the eigenvalues are distinct and come in conjugate pairs (except for the Perron eigenvalue). My intuition is that the initial vector will have a single nonzero entry. An example of a Leslie matrix we are using is L = 0 0 0 9.023 48.779 0.0582 0 0 0 0 0 0.8 0 0 0 0 0 0.8 0 0 0 0 0 0.652 0 Which is "close" to an imprimitive matrix since the birth rate in the last age group is much greater than the others. This means that population waves (oscillations) will be important. We can show that the population trajectory is v(t) = w1*c1*lambda1^t +(r2)^t*{cos(theta2*t)(c2*w2+c3*w3)+isin(theta2*t)*(c2*w2-c3*w3)} +(r4)^t*{cos(theta4*t)(c4*w4+c5*w5)+isin(theta4*t)*(c4*w4-c5*w5)} where v(0) = c1*w1 + c2*w2 + c3*w3 + c4*w4+c5*w5 (such a linear combination of the eigenvectors with weights c1, c2, ..., c5 is possible because the e.vectors are linearly independent). In the above, lambda1 is the Perron eigenvalue, r2 is the modulus of the second and third eigenvalues, and r4 is the moduls of the fourth and fifth eigenvalues. w1,w2,w3,w4,w5 are eigenvectors corresponding to the five distinct eigenvalues, such that w2=Conj(w3), w4=Conj(w5), and the weights are such that c2=Conj(c3), c4=Conj(c5), and c1 is positive and real. (w1 is real and can be chosen to have positive entries). theta2 is the the angle that the 2nd eigenvalue makes with the positive x axis in the complex plane, and theta4 is the angle of the 4th eigenvalue. The eigenvalues are such that lambda2=Conj(lambda3) and lambda4=Conj(lambda5). The problem is to choose v(0) to excite the greatest population waves in the salmon population (waves of maximum amplitude), given that ||v(0)||=1, where ||.|| is the l2 (Euclidean) norm. The waves are associated with the terms in the {} in the above equation.(Note that v(0), the initial vector, must be nonnegative). Are there general rules for choosing v(0) based on the entries of L? For discussion of population waves, see Bernadelli (1941). Thanks in advance. Reference: Leslie, P.H. 1945. On the use of matrices in certain population mathematics. Biometrika 33:183-212. Bernadelli, H. 1941. Population waves. Journal of Burma Research Society 31: 1-18. -- Richard A. Hinrichsen Lakeview Medical Dental Building 3216 NE 45TH PLE STE 303W Seattle, WA 98105-4028