Nowak project 1. Linear case: xi' = (1-q)Er., a„x„ - (qa, + di)x, xfig = - Olan + xn- y'= by -dy. Eigenvalue condition for the x equation: 1= Tf n qak nXi (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 clak > 1 . q 4d.ai rik=l(b-d+ Nak + dk (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 L V 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 Lrrn k I iza Lk.' (IC +1) - 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 E ( f (l+t)2441 ti-t r dt • 0 (1.8) The sum in the integrand is geometric, and what is written above is equal 11 sdt - f lr0+q4.4 0+0..dt • 0 (1.9) The right hand integral is indeed equal to 11_ 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 Eisatinciii+ + ((qaqa+b))mIrkzo((qa q:b))k (1.10) Evaluating these sums gives the instability condition caq:b)m-I ((q+q „r q -I • 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-sz fr q(cok -c1) qAu kml(qco+b)k +(bc2 -qci) — 1 • (1.12) This can be rewritten as = 1 , (1.13) (leo ci d (4— bc2 '4`' The n'th term in the sum in (1.13) can be where y = cietp0 = an v qco + b • written as tn-Ct Fr 0+00.4 dt where p = j t-a04P dt . (1+ (1.14) This understood, interchange the integral with the sum to rewrite the sum in (1.13) as ir t +titt- 1+a 7P J -cnzo(ar dt = 0 (1 (0 -100 dt. ci 0+04 (1.15) The stability condition in (1.12) can be restated as — t -a,„ dt . qcg oc°+b J0 (i-Ft9(1+(1-y)t)dt > 0 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 • .1t-a0+0,4 dt= BOx+(3,1-a) Fa3,a+I3 ; 1+(i;0) . (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 >1q .1.12111k.1 (qak + b) (1.18) Setting ak = (qacika+1 b) , this is equivalent to the condition that a, + ay; + ala2a3 + • > 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, al is the probability of having x2 given xi, then ccia2 is the probability of x3 given ; and alcc2a5 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