Nowak project 1. Linear case: xis = (1 a„x„ - (qa, + dl)x, x„' = - (qa„ + daxo. y'. by- dy. Eigenvalue condition for the x equation: - _ 1g Gin?, 11,4 ( 4•4 .do) Note that X > 0 requires that pt • n q (cial:1 -41t) zI (1.2) The condition X> b-d is needed for growth faster than that of y. This condition reads n qak g q n >1 .fia2111.4.1 +(qak + (1.3) In the case when ak = a and dk = d is constant, then the condition in (1.1) asserts that 1 = a Ems, II with ri = qa(X + qa + d)'i. This is to say that ATI =MI -11) and so ri = 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 z* = 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 z* = I . a) Neutrality Martin suggests considering the case dk = d in which case the condition X = b- d reads I - q v rin qak — hdezi A M., NaL + b) ' (1.4) Martin claims that this condition is obeyed if a, = k b. In the latter case, the condition in (1.4) reads EFTA_R1_0 1997757 EFTA02682655

v ri• Li•zi 1 1 L-I (9k + - • (1.5) To verify that this is indeed the case, introduce for the moment 11 to denote 1/q. What is written in (1.5) is equivalent to the assertion that Eanne k kti (k+i) q - I • (1.6) A given term in this sum is equal to f in J (1+00+91dt . 0 (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 1 e— J00.I.cp+n nal I+lr dt (1.8) The sum in the integrand is geometric, and what is written above is equal ri t dt = f dt 0 (1+01+11 (1.9) The right hand integral is indeed equal to II tt . b) ak =bfork<mandat =afork>m Martin asks for the case ak = b for k < m and at = a for k m with a > b. I assume again that all dt = d. In this case, the left hand side of (1.4) reads 9 v Leisn ( N+ 41I))11 ( 41:1:137LOV II+0, (1.10) Evaluating these sums gives the instability condition r -i I (Sri > (qa+b) b (q+ I) I -q ' c) ak is a rational function of k EFTA_R1_0 1997758 EFTA02682656

The next case Martin asks about is that where ak = (cok - c,)/(k + c2) where the constants are chose so that b = (c0 - 0(1 + c2). The neutrality condtion in (1.4) reads v n • q(cok - ) q ilk -I (qc0+b)k + (bc2 qci ) = 1 ' (1.12) This can be rewritten as grk.lm = 1 Laiy (1.13) where y= —qc° , a = , and (3 = . The n'th term in the sum in (1.13) can be goo.o qco an + b written as lap e-ct 0+00+,4 dt where p - dt (I tyP (1.14) This understood, interchange the integral with the sum to rewrite the sum in (1.13) as yt t-a TV 1 0+t-t;.,,8 L oao(M)" dt = y p'' (1+0130+o-110dt . (1.15) The stability condition in (1.12) can be restated as qco f t-a dt cico+b 0+00(1+010dt > 1-q 0 0+014 (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 de • dt = (1- yr IB(a+p, I-a) F((i,a+I3 ; I +13;y). 0+430+0- • dt = B(a+ 13, I-a) F(I3,a+ ; I +13;0) . 0+04 (1.17) EFTA_R1_0 1997759 EFTA02682657

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 - q v rr . qak Lem!. 1U.! Nat +b) > I. (1.18) gat Setting a t — Kok 4. b ) this is equivalent to the condition that a, + a,a2 + ala2a3 + > . (1.19) What follows is a thought about an interpretation: Looking at the equation for xbi, I can think of at., as the probability of creating some ; given This understood, a, is the probability of having x2 given xi, then ala2 is the probability of x2 given x, and ala2a2 is the probability of ; given x„ 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. EFTA_R1_01997760 EFTA02682658