Biol 54H
Nitin Sekar, Tim Alves, Michelle Lee, Katy Flinn, and Thanveer Gadwal
Dynamics of Cannibalism
Introduction:
Tiger salamander larvae all have the potential to take on two forms: an omnivore
morph and a cannibal morph. The cannibal morph has a larger head than the omnivore
morph, and prey on the smaller, omnivore morphs. Cannibal morphs, however, avoid
eating their kin. In order to observe some of the dynamics of the interactions within a
population of these salamanders, we have simplified the situation in a way that could be
represented in our model. One such simplification is the creation of two sexually
exclusive families to making discerning relatedness easier. Family A consists of all
individuals with AA, Aa, and aa genotypes; family B consists of individuals with BB, Bb,
and bb genotyped individuals. The family with the minority population will display a
propensity to become a cannibal that is determined by their genotype. The other,
majority family will consist of only omnivores.
To make this clear in our model, we have created a threshold of family frequency
at 0.5. If a family’s consists of more than half the salamander population (in other words,
if the frequency surpasses the threshold), that family will not show cannibalism because
the majority of individuals in the population are related. At the threshold, we arbitrarily
decided that family B would show cannibalism, and that A’s would remain omnivore
morphs. This way only one family will have cannibals at any given time. Cannibals will
not eat individuals from their own family. Our gene for cannibalistic propensity is A for
family A and B for family B. It should be understood that these becoming a cannibal is
the result of both genetic and environmental cues: an individual needs to be in the
minority family as well as have the genes allowing the development of cannibalism. Even
then, only a certain proportion of the individuals with that genotype will become
cannibals. This proportion varies for different genotypes; AA individuals will have a
different probability of becoming a cannibal than Aa or aa individuals. Here we realized
that the effects of each allele could be either additive or totally dominant/recessive. We
experimented with this in our model.
Both the omnivore and cannibal morphs face a fitness loss. Cannibal morphs
devote more energy to gaining size and finding food, and are thus at a slight selection
disadvantage. Omnivore morphs may have decreased fitness due to predation by the
opposing family’s cannibal morphs. Omnivore morphs in the family displaying
cannibalism, however, have a fitness of 1 for that generation.
In our model we have also included migration. A neighboring lake has become
fixed for the AA individuals of the A family. These individuals are all non-cannibalistic
during migration since only family members are present in their lake of origin. Since they
are capable of migrating to our population, their morph has already been determined- but
their offspring will face the same probability of becoming cannibals as the locals.
We wanted to model this population because so many parameters determine
which family goes to fixation. It includes kin selection, sexual selection, and migration.
Having created this dynamic model, we established three questions to answer:
1. Which factors contribute to the success of one family over another?
2. Under which conditions can the two families coexist indefinitely?
3. How do levels of cannibalism compare to allele frequencies?
We altered the threshold “winner,” starting frequencies, type of dominance and
migration rate to help us answer these questions.
MODEL
Given the goals stated above, this model will track allele and genotype
frequencies at one locus, as well as the frequencies of the two possible phenotypes based
on this locus for each family over many generations. There are two families: family A,
which has alleles A and a at this locus, and family B, which has alleles B and b. The
model will track the frequencies of the A, a, B, and b alleles as they change over time.
Given these alleles, and the fact that the two families are sexually exclusive, the possible
genotypes are AA, Aa, aa, BB, Bb, and bb, which will also be monitored over time (but
not graphed). Phenotyping in this model consists of one character: cannibalism.
Individuals are either cannibals (denoted ‘E’) or non-cannibals (denoted ‘e’). Possible
phenotypes are therefore EAA, eAA, EAa, eAa, Eaa, eaa, EBB, eBB, EBb, eBb, Ebb, and
ebb.
The aforementioned frequencies depend on the five life stages in this model.
Stage one is birth. Each salamander is born diploid, and the morph of the individualcannibal or omnivore- has not yet been determined. Stage two consists of determination
of cannibalism. Whether a particular salamander will develop into the cannibalistic
morph depends on two things: a) the genetic predisposition of the individual to become a
cannibal, determined by the single locus we are tracking, and b) the proportion of nonfamily members in the population. This genetic predisposition, given as a probability, c,
depends on the particular genotype (each genotype has a different c value- e.g., cAA for
genotype AA), and part of the model will look at how different types of allele interactions
(dominance, overdominance, etc.) affect the population. Also, since the tendency to
become a cannibal depends on the population density and degree of relatedness of
surrounding salamanders, a threshold was included in the model to determine which
family would have the potential to become cannibals. If a family’s (A or B) overall
frequency is less than 0.5, then this population will express its genetic predisposition to
cannibalism (based on probabilities associated with genotype). At the threshold point of
0.5, one family is arbitrarily given the ability to become cannibals. Note that only one
family at a time can become cannibals, so while the A family may become cannibals the
B family can not and vice versa. A particular goal of this model is to look at the impact
of this threshold on the population as a whole. Stage three is natural selection. In the
model, the non-cannibalistic salamander family faces a selection coefficient, s, which is
dependent on the frequency of cannibals in the other family. This is because a major
assumption of the model is that cannibals of one family will only feed on salamanders of
the other family (non-related). The cannibalistic family will face a selection coefficient,
u, which is dependent on the particular genotypes’ probability of becoming a cannibal
(c). This is because the model assumes that cannibals have a unique decrease in fitness
because of higher energies needs associated with growing larger bodies capable of
cannibalism. Also, by multiplying u by the c for a particular genotype, we effectively
only reduce the fitness for cannibals of that genotype, not non-cannibals. Stage four of
the model is migration, which consists only of immigration of salamanders of the AA
genotype into the current model’s population. These immigrants are assumed to be noncannibals (because their original population has only members of the A family, so
cannibalism cannot develop). These non-cannibal AA salamanders immigrate at a rate
‘m’ which will be studied as one possible factor that can heavily influence population
dynamics. The fifth and final stage is mating, which is considered to be random within a
family (remember that A’s will not mate with B’s and vice versa). A slightly modified
version of the Hardy-Weinberg equation is used to determine allele and genotype
frequencies of the resulting offspring. The life cycle has thus come full circle and now
will repeat over many generations.
Now that the model is complete and ready for simulation, a standard
state/condition must be created as a starting point for further testing and variation of
parameters. The starting allele frequencies in the standard state are all equal, that is pA,
pa, pB, and pb are all equal to 0.25. The standard threshold is t=0.5, and if the proportion
of the A family is less than 0.5, they will have a propensity to become cannibals and if
the proportion of the B family is less than or equal to 0.5, they will have the
predisposition to cannibalism. That is, at the threshold value t=0.5, the B family is
arbitrarily given the ability to become cannibals. The standard selection coefficients are
s=0.5 and u=0.1. These parameters will not be varied in future testing. The standard
migration rate is m=0- that is, there is no migration. The standard state for genetic
predisposition towards cannibalism, particular to each genotype, is as follows: cAA=0.7,
cAa=0.35, caa=0, cBB=0.7, cBb=0.35, cbb=0. Please note however, that these values
only apply if the threshold criteria is met, otherwise they are all 0 for the noncannibalistic family.
Except for the t, s, and u, we vary all these standards to see the effects on the
population.
RESULTS
Testing the thresholdUsing the standard conditions given above and varying which family becomes
cannibals at the threshold point t=0.5, the model shows that the family that becomes
cannibalistic at the threshold point fixes while the other family dies out. Since the model
starts out with equal starting frequencies, the proportion of each family in the initial
generation is exactly 0.5, the threshold point. Thus, all else being equal, the family that
starts out as cannibals will overcome the other family, which goes extinct locally.
Testing initial starting frequencies
All else being equal, the model shows that the family that begins with a higher
proportion of individuals survives, while the other family dies out. This applies in cases
where pB=pb; an exception can be found when pB > pb and pA< pa, even when
pB+pb>pA+pa.
In addition, keeping each family at half the population, we manipulated the
starting frequencies of A and a. When A began at 0.4 and a at 0.1, the A family survived
and the B family went extinct. However, when A started at 0.26 and a at 0.24, the B
family survived due to their advantage at the threshold, and the A family died off.
In this regard, the model also shows that it is the frequencies of the genotypes that
carry a tendency to become a cannibal have more influence on the fate of the population
than the frequencies of those genotypes that will never become cannibals (aa and bb).
Testing migration
Based on standard conditions, the B family will dominate the population and the
A family will die out. It was therefore hypothesized that migration of AA individuals
could change the outcome. After extensive testing, the group found that somewhere
between m=0.00138300 and m=0.00138325 is the point where the migration rate can
balance out the effect of B’s advantage in receiving the cannibalistic tendency at the
threshold. If migration is less than or equal to 0.001383, the B family will still win out.
If m is greater than or equal to 0.00138325, the A family will take over the population.
Even such a small rate of migration can quickly turn the tides against the B family.
Discussion:
Altering various aspects of the model gave us various insights into the dynamics of
the population.
The Effects of the Threshold:
Varying possession of the threshold underscored what we learned in evolutionary
biology: that if everything else is equal, the smallest advantage can lead a population or,
here, a family to evolutionary triumph. Here, just the ability of a family to become
cannibals at family frequency 0.5 gave that family the push it needed to out-compete the
opposing family.
In a way, this also proves the imperfection of our model- since the threshold is always
in possession of one family or the other, we cannot necessarily test whether or not any
other single factor is enough to give a family an adequate advantage to reach fixation. For
example, with migration, the immigration of A individuals has to be large enough to
overcome the advantages lent to the B family by possession of the threshold. Thus, in a
way, the rest of our model ultimately tests what is necessary to overcome the threshold
advantage.
Varying the Starting Frequency:
If pA=pa and pB=pb, and we change nothing but the starting frequency of our
population, the population that begins with the higher frequency will go to fixation.
Regardless of which population has the propensity for cannibalism at the threshold, the
higher population will achieve fixation. This suggests that even though the population at
the lower frequency will exhibit cannibalism, the higher population numbers will allow
the family to succeed. The fitness decline felt by the non-cannibalistic family due to
predation is not high enough to wipe them out. Cannibalism, therefore, is not an allpowerful evolutionary development! It is of great help at the threshold, but in this
scenario is relatively unhelpful to the family.
Varying Dominance:
Here we experimented with changing between additive and dominant/recessive
traits. The family within which the cannibalistic tendency exhibited incomplete
dominance (ex. cAA=0.7 and cAa=0.35) consistently went to fixation over the family
where the cannibalistic tendency was recessive, To see if the sum of probabilities affects
which one goes to fixation we changed the probabilities to sum to 0.95 for A, and 1 for B
(meaning that cBB = 1). In this case family B survived for amore generations, but family
A still went to fixation. Since the additive effects of alleles A and a overcame even the
deterministic BB genotype (all BB individuals became cannibals, we conclude that
additive effects have a very strong effect on ability to survive in this population.
Varying Migration:
In varying the migration rate we found that the number of generations in which
both families survive was much longer. We quested for the magic migration rate that
could enable both families to live in an oscillating state of cannibalism and noncannibalism forever, and after going to past 7 decimal places gave up. What this shows
us is that in a model as complex as this, finding an equilibrium by trial and error is
extremely difficult.
Cannibalism Overall:
As noted before, a greater tendency for cannibalism is not always enough to help
a family overcome odds in initial frequencies, incomplete dominance, etc. In fact,
comparing trends of cannibalism of families to allele frequencies indicates that sustained
cannibalism is actually a forerunner to extinction. As a family’s spikes of cannibalism
become smaller and smaller, the observer can be assured of that family’s success.
Conclusions:
First of all, the existence of a set number in our model, the threshold, eliminated
any chance of our finding the equilibrium frequencies of the alleles using what we know
of mathematica. In a way, this showed us that in complex circumstances, we can only
observe trends, not always come to concrete conclusions.
This is reiterated by the experiments changing ratios of starting alleles (e.g.
pA=0.4, pa=0.1 v. pA=0.26, pa=0.24) and our observations of migration rates. Our
inability to come up with a clear cut rule for when each family would triumph shows our
inability to come up with definite answers due to the complexity of a model, where so
many things may vary at once. We were often unable to simplify the model enough to
decide the absolute importance of certain factors. Thus, perhaps the greatest lesson in all
this is that we are not near ready enough to model real situations, which are much more
complex than this.
Work Cited:
Pfennig, David. “Kinship affects morphogenesis in cannibalistic salamanders. Nature.
362, 836 – 838.