Struggle for Existence
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© 1997
David H.A. Fitch
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Lecture notes

Darwin's mechanism of evolution:  "Struggle for existence"

I.  What is meant by "struggle for existence"?

A. "Struggle" is metaphorical and anthropomorphic:  implies competition to survive and reproduce (which is the more important)

B. "Struggle" results from the tendency of ALL organisms to produce more than can actually be supported
1. Principle of Malthus (if unchecked, the human population will increase indefinitely and famine will result), applied to all life
2. That is, they tend toward a geometrical (exponential) increase:

  1. Per capita rate of change in population number, r, is determined byper capita birth (b) and death (d) rates:
      r = b - d
  2. Rate of change in the number of individuals in a population:

Apply per capita rates to numbers of individuals in the population (N).

Multiply each side by 1/N.

Integrate both sides.




  1. To determine population size (N) at time t, assume that r = rm, a constant (i.e., the "intrinsic rate of increase".  Then:

               ln(N) = rt + c

               N = e(rt+c) = ertec = ertc ;            N = cert

               N0 = cer*0 = c * 1 = c;             c = N0

               Nt = cert = N0ert;                   Nt = N0ert

C.  This "geometric increase" is not just theoretical, but empirical (Darwin uses documented examples, but makes math errors in the first edition that are corrected later)

D.  Because of this tendency toward geometric increase, this increase must be checked by destruction at some point.

II.  For each population, we can describe an "intrinsic rate of increase" (i.e., potential rate of increase assuming no limiting factors), rm, which is dependent on 2 population-specific factors:

A.  Survivorship and Fecundity
(Also the two factors that help define an individual's "Fitness")  These can be worked into the equation describing a population's growth rate as follows:

  1. From earlier, Nt = N0ert
  2. The proportional increase in the size of a population (i.e., the ratio between the sizes of successive generations, assuming no change in r from generation to generation) is:

    Nt+1/Nt , which is equivalent to Nt/N0 = ert .
  3. This proportional increase is due to survivorship (i.e., the proportion of mothers that survive to reproduce) and fecundity (i.e., the number of female offspring), over all ages (x):

(L is the eldest member of the population.)

Divide each side by ert.

B.  This equation describes density-independent growth (i.e., where r = rm) such that:
1.  Resources are unlimited
2.  No predators
3.  Stable age distribution

III.  Density-dependent growth

A.  But actual, observed rates of growth (r) are usually 0 or at least not constant, due to natural forces that limit population growth:
1.  Possible synchronous population replacements
2.  Genetic differences between individuals (i.e., individual variation in rm)
3.  Age distribution is not stable
4.  Birth and death rates are different for each age group
5.  Accessibility to mates may be imposed by social interactions, etc.
6.  Growth may be limited by environment

B.  Darwin writes that specific causes of mortality are often not known, but probably:
1.  Eggs or very young suffer most
2.  Mortality factors may be:
a.  Limitation of food and other resources
b.  Prdators
c.  Climate
d.  Disease and parasites, esp. in dense populations
e.  Population size itself

C.  To describe such limitations on population growth, we use the LOGISTIC EQUATION (a model based on the logic of what should happen to a population, given an upper limit to population number:  the carrying capacity, K, which is simply the maximum number of individuals that can exist in a particular population):

  • When the niche is empty, N<<K and N/K is small; rate of growth is maximal, and approaches the intrinsic rate of growth, rm
  • When the niche is nearly filled, N is almost K, and N/K approaches 1; then the rate of growth is small, and approaches 0.

IV.  Ecological context also defines the "Struggle"
Population dynamics in Nature are very complex because they depend on Interactions with other individuals and species.  Thus (p. 73), cataclysms and miracles need not be invoked to explain results of complex interactions.  Most importantly:

The "Struggle" is probably most severe between members of the SAME population (and/or similar varieties or species)

The LOGISTIC EQUATION can be used to test the hypothesis that COMPETITION can result in REPLACEMENT of one kind of organism by another:

A.  If each utilizes different resources and they do not compete in the same niche, each population will attain its own carrying capacity independent of the other population.

B.   But assume they are limited by exactly the same resources in the same niche (i.e., N1 is slowed down by N2's growth as much as N2 is by its own growth):

Population 1:


Population 2:

  • If K1 > K2 (e.g., if individuals in Population 1 are smaller and need less food), then N1 and N2 increase until N1 + N2 = K2.
  • Eventually, Population 2 (N2) is replaced by Population 1 (N1)!

C.  Conclusions from this thought experiment:

Assuming that a population's growth is limited (i.e., there is a "carrying capacity", and therefore assuming the logistic equation)...

1.  When different populations are competing for the same resources, the population with the greater carrying capacity (or intrinsic rate of growth) will replace the other.

2.  Coexistence can occur only if each kind of population has a greater inhibiting effect on its own growth than on that of the other kind (e.g., if the 2 kinds compete for some types of food, but there are also kind-specific resources).

3.  When there is coexistence, each kind has a higher rate of population growth than its competitor when it is rare and its competitor is common (and thus near its K) (i.e., there is frequency-dependent selection).

(Because the word "species" is just Latin for "kind", substitute the word species into the above arguments.)

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  1. Using the logistic equation for two populations that share resources, show how one variation can be replaced by a new variation that affects r, the rate of growth (e.g., if the new variant becomes sexually mature at a younger age).
  2. Why is replacement of one variation by another an important part of Darwin's general theory of the mechanism of evolutionary divergence?

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Struggle  Natural
[The Struggle] [Natural Selection]


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