Nowak project
1. Linear case:
xis = (1 a„x„ - (qa, + dl)x,
x„' = - (qa„ + daxo.
y'. by- dy.
Eigenvalue condition for the x equation:
_
- g1 Gin?, 11,4 ( 4•4 .do)
Note that X > 0 requires that
pt • n
q -41t) zI
(cial:1
(1.2)
The condition X> b-d is needed for growth faster than that of y. This condition reads
n qak n >1
g q .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
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v ri•
Li•zi1 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
Eannekti (k+i)
k
q-I •
(1.6)
A given term in this sum is equal to
f in dt .
J0 (1+00+91
(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 J00.I.cp+n nale—
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
v
9 Leisn ( N+
41 I) )11 ( 41:1:13 7LOV +0,
II
(1.10)
Evaluating these sums gives the instability condition
(qa+b)
r -i Ib (Sri
(q+ I)
>
I -q '
c) ak is a rational function of k
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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
Laiygrk.lm = 1
(1.13)
where y= —qc° , a = qco , and (3 =
goo.o . The n'th term in the sum in (1.13) can be
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-110 dt .
(1.15)
The stability condition in (1.12) can be restated as
qco f 0+00(1+010dt
t-a > dt
cico+b 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 I B(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)
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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.
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