ratios? Hypotheses can be proposed, and since relative numbers of distinctly different
types of individuals are involved, the feasibility (not the reality!) of those hypotheses
can be tested by numerical modeling. The strong prevalence of males maturing in
some field observations, combined with the certainty that males have much higher
field mortality rates than females (males also mature earlier), suggests that an
advantage should accrue to stocks with many more males at maturation than females.
That can be tested with an IBM.
(^) Let = No. of young maturing as females, = No. as males and T = + . From
simple encounter theory, the expected number (or rate) of male–female encounters (E)
for mating will be proportional to the product of their abundances: E = C , in which
C is an “encounter kernel” (see Chapter 1). Then, since = T − , E = C (T − )
and E = CT − C^2 . Thus, the number of encounters should have a parabolic
relationship to the relative numbers of males and females. Also, at least with no
differential mortality, since setting dE/d = CT − 2 C = 0, we find the number of
males (and females) for maximum E is = T/2 (i.e. equal abundance of the sexes).
However, those are not necessarily the relative proportions generating the most
fertilized eggs (a function of fertilized females), especially if there is much greater
post-maturation mortality for males. There is doubtless an analytical approach, but
some answers are readily found with an IBM, also recognizable as a Monte Carlo test.
The process can be grasped by study of the program in Box 4.5. It calculates the
number of mating encounters among 20,000 adults (increase that to get greater
precision; reduce it to make the program finish sooner) over 45 “days” (arbitrary time
steps) with an arbitrary value of C and sex ratios varying from 10% to 90% males.
The likelihood of encounter at each time step for each male with each female (a pair-
by-pair equivalent of C) is set very low, which would be correct for almost any
number of adults in a sizeable search volume. When a female is mated twice she is
removed from the female matrix, likely a realistic feature of mating in Calanus.
Box 4.5 Program for an individual-based model of
variation in copepod mate-encounter rates as a
function of adult sex ratio
(^) This takes a while to run. Take a walk in the sunshine, then check.
(^)
%Copepod-mating encounter-rate model
%This plots total matings as a function of proportion of males.
%It also counts matings/male (no limit) & matings/female
%(the latter limited to 2).
T=20000; %Total adults in population
C=0.0000045; %c=small daily probability a specific male will mate
%with a specific female
%Prepare some storage:
A=zeros(9,1); Propmales=A; Storem1=A; Storem2=A; Storef1=A; Storef2=A; MpM=zeros(9,8);
for h=1:9 %Loop over proportions of males from 0.1 to 0.9
m=2000*h; f=T-m; Propmales(h)=m/20000;