EFTA01113651.pdf

Nowak project 1. Linear case: xi ' = (1-q)E r., a„x„ - (qa, + di)x, xfig = - Olan + xn- y'= by -dy. Eigenvalue condition for the x equation: 1= nXi Tf n qak (qak + dk )) Note that X > 0 requires that v n. clak I q 1 I k=I (qak +4) (1.2) The condition A.> b-d is needed for growth faster than that of y. This condition reads I -q v q 4d.ai rik=l(b-d+Nak clak + dk > 1 . (1.3) In the case when ak = a and dk = d is constant, then the condition in (1.1) asserts that 1 = Lag En,11 with n= qa(X + qa + d)". This is to say that 1-3— = -ri) and so n -q = q. Thus, X + qa+ d = 2qa and so X = (1 -q)a - d. Growth faster than the y-model requires (I -q)a > b which is maybe expected. Martins `system with food' on page 2 at equilibrium e = d/b gives the linear instability condition that is identical to (1.2) with the replacement q z*q. This understood, I will address the remaining questions on the bottom of page 2 with e = 1. a) Neutrality Martin suggests considering the case dk = d in which case the condition X = b- d reads v n. qak = ' iLinZi I (qak + b) (1.4) Martin claims that this condition is obeyed if ak = k b. In the latter case, the condition in (1.4) reads EFTA01113651  • q V L ind n (qk+ I) - , (1.5) To verify that this is indeed the case, introduce for the moment rt to denote 1/q. What is written in (1.5) is equivalent to the assertion that Lizarrn (IC +1) k Lk.' I - 11 -I (1.6) A given term in this sum is equal to 11rr to dt . Jo (ti-tr+1+1 (1.7) as can be seen using n successive integration by parts. This being the case, interchange the sum and the integral. The result on the left side of (1.6) is then TI f (l+t)2441 E ( ti-t r dt • 0 (1.8) The sum in the integrand is geometric, and what is written above is equal 11 sdt 0+q4.4 - f 0+0..dt lr • 0 (1.9) The right hand integral is indeed equal to 1 1_ b) ak = b for k < m and ak = a for k > m Martin asks for the case ak = b fork < m and ak = a for k a m with a > b. I assume again that all dk = d. In this case, the left hand side of (1.4) reads t (14 E isatin ci + ii + ((qaqa +b))m Irkzo((qaq:b)) k (1.10) Evaluating these sums gives the instability condition caq:b)m-I ((q+q„r-I q• c) ak is a rational function of k EFTA01113652  The next case Martin asks about is that where ak = (cok - c,)/(k + c2) where the constants are chose so that b = (co - cl)/(1 + c2). The neutrality condtion in (1.4) reads L-qsz fr Au q(cok -c1 ) kml(qco +b)k +(bc2 -qci ) — 1 • (1.12) This can be rewritten as = 1 , (1.13) (leo where y = cietp0 = ci and (4— v bc2 ' 4`' qco + b • The n'th term in the sum in (1.13) can be written as tn- Ct Fr 0+00. 4 dt where p = j t-a dt . (1+04P (1.14) This understood, interchange the integral with the sum to rewrite the sum in (1.13) as ir t +titt-a dt. 7P J -c nzo(ar dt = 0 (1 (1+0 -100 ci 0+04 (1.15) The stability condition in (1.12) can be restated as (i-Ft9(1+(1-y)t)dt > 0 — t-a,„ dt . qcgoc°+b J0 q 0 (1+t)rF (1.16) According to Gradshteyn and Ryzhik, (Tables of integrals, series and products; Enlarged edition, I. S. Gradshteyn and I. M. Rhyzik; Academic Press 1980), these definite integrals can be expressed in terms of two special functions, these denoted by B (this being the 'beta function' or `Euler's integral of the first kind') and F (this being `Gauss' hypergeometric function'). In particular, Equation 9 in Section 3.197 writes CO • -1 at-a dt -(I yrrri B(a+ (3,1-a) F03,0(4(3;1+13;10. 0 0+01.(1+011 ) OD t-a dt= BOx+(3,1-a) Fa3,a+I3 ; 1+(i;0) . • .10+0,4 (1.17) EFTA01113653 For what it is worth, the special functions B and F are defined respectively in Sections 8.38 and 9.10-13 of Gradshteyn and Ryzhik. d) Interpreting the instability condition Martin asks for the meaning of the condition that I -9 v n. Sac >1 q . 1.12111 k.1 (qak + b) (1.18) Setting a k = (qacika+1 b) , this is equivalent to the condition that a, + ay; + a la 2a 3 + • > ATI . (1.19) What follows is a thought about an interpretation: Looking at the equation for xkm, I can think of cc., as the probability of creating some xk given xk.I. This understood, a l is the probability of having x2 given xi, then ccia 2 is the probability of x3 given ; and a lcc2a 5 is the probability of ; given xi, etc. The sum on the right can be thought of as a sum of conditional probabilities. I shall think more about this as a path to an interpretation of (1.19). e) Other forms of density regulation I haven't had time to consider these yet. EFTA01113654