Names Problem Set: Population Size and Competition Previously, we looked at population dynamics. Now let's venture into community ecology, specifically competition between two species. Interspecific competition is a symbiosis between two species such that the presence of either species represents a cost to the other. Four terms that you will need to know are: conspecific = member of the same species; intraspecific = within the same species; heterospecific = member of another species interspecific = between different species Let's get started! 1. Name two species that are competitors in nature. (Neither species can be lion or hyena.) To model how populations are affected by the presence of a competitor, let's review the logistic growth equation: dN where dN dt =rN TN (K - N) K - is the change in population size (N) at a given time (t). r is the intrinsic rate of growth and K is the carrying capacity. dt What we are interested in is the per capita increase. That is, we are interested in the rate of change in population size per each individual in that population. 2. How would you modify the above equation to make it per capita? Now let's assume that population size is quite small, and K-N K capita rate of increase is essentially equal to (4.) exponentially. is very close to what value? (3.) Therefore, when N is quite small, the population increases So the per But let's assume the change in per capita growth rate is linear and that with the addition of each individual, the per capita growth rate decreases. Think of it this way. When a conspecific is added to a population, the per capita rate of increase of that species decreases a certain amount. This decrease is a result of intraspecific competition (i.e., competition among members of the same species). For example, if you have a population of 20 lions they each get a 5% share of the food. If you add a 21st lion to the population each lion will now get a 4.7% share of the food and less food should lead to slower growth and possibly fewer offspring. But how does that translate into a change in the per capita growth rate? Think of another population where each new individual decreases the rate of growth by 0.02. You can think that for every individual added to a population, the average reproduction rate decreases by 0.02. individuals. 5. Here is another approach. Imagine a population of 100 parthenogenic fish. Parthenogenic fish are females that produce clone offspring without sex. In this parthenogenic population of all females, the per capita decrease in growth rate is 0.02, as above. Each additional fish decreases the clutch size by 0.02. The average clutch size for each female is 4. If there are 150 fish the next year, what would the average clutch size be? (Think about how many more fish there are and how this number will affect the reproductive rate.) 5A. As an aside, think about the number of offspring each year. How many offspring are produced when there are 100 fish? How many when there are 150? So the per capita rate of change has decreased (see previous answer), but has the population's rate of change increased or decreased? So how do we incorporate this decrease into our equation? Essentially it acts like a correction factor. So remember that for now N is low we can ignore K and we modify the equation thusly: dN Ndt =r-0.02N Now consider a second species that acts as a competitor (species 2). If each member of species 2 decreases the per capita growth rate of species 1 by 0.005, we can further modify the equation by: dN, = r₁ −0.02N₁ −0.005N2 N₁dt So you can see that the per capita growth rate of species 1 is affected by its own r, its own population size, and the population size of its competitor. Note how we use the subscripts to indicate the species. There are now two factors that decrease per capita growth rate: intraspecific and interspecific competition. To make two general equations: dN₁ N₁dt - dN2 N₂dt 1 = √2 - α 22 №2 - a₂₁N₁ where the first subscript is the species being affected, the second subscript is the affecting species. a12 is the effect of competition on species 1 from species 2. POPULATION EQULIBRIUM Population equilibrium is when there is no change in population size: things are stable. This is a very interesting point in ecology dN = 0. The equilibrium point of each population will depend on its r and the size Ndt and a good place to start. At equilibrium, of its population and its competitor's population. 6. Use the general equations above to determine the equilibrium population size for species 1 (i.e., N₁) when species 2 is absent. (You won't solve for a specific value, but for a function. Remember there will be many equilibria possible depending on the situations.) 7. What is the equilibrium population size for species 2 when species 1 is absent? So, essentially a species will be stable when its population size is equal to the ratio of its intrinsic rate of growth over the cost of intraspecific competition. That makes some sense, although it doesn't seem terribly obvious. Let's think of it another way. Before we continue with multiple species, think of a single species with a constant intrinsic rate of growth. In habitat A the cost of intraspecific competition is higher (i.e., each new conspecific affects the other conspecifics more) than in habitat B. 8. Complete this table for the above scenario. N is the equilibrium population size. In the middle column, use one of these three symbols: >, < or =. Habitat A r a rla N Habitat B r a rla N 9. So when intraspecific competition is stronger in one habitat than another but all else is equal, what can you say about the sizes of the populations in those two habitats? Now think of a species with a varying intrinsic rate of growth, but a constant cost of intraspecific competition. In habitat A the intrinsic rate of growth is higher than in habitat B. 10. Complete this table for the above scenario. A r a r/a N B r a r/a N Make sense now? When a population has reached the point at which the environment can no longer support any further increases and population size becomes stable, what do we call that value? The carrying capacity (K). So you have just derived a different way to look at K. Rather than a straight number, it's a ratio of two numbers. We have derived the K of a population without any interspecific competition. INTERSPECIFIC COMPETITION Awesome! But to have interspecific competition you have to add another species. That's the whole point. And to look at the strength of the competition, you cannot just look at the effect an individual of one species has on another. The best way to look at it is by measuring the interspecific effect relative to the intraspecific effect. Essentially, the strength of competition is how strongly a second species affects the first relative to how much the first affects itself. Now imagine we have two species (1 and 2) with the following: a11 = 0.06, a12= = 0.002 = a22 0.08, a21 = 0.04 11. For each individual of species 2 added to the environment, by how much is the growth rate of species 1 decreased? What about the growth rate of species 2? 12. Do both species affect each other equally per capita (i.e., how 1 affects 2 vs. how 2 affects 1, not to be confused with the last question)? If not, which one affects its competitor more? Remember, a species with a small a might have a greater overall effect on a second species if there are enough individuals. For example, if each individual hyena has a small effect on lions but there are millions of hyenas, then the effect on lions would still be strong. It's just that the per capita effect would not be strong. So one facet of interspecific competition is what is often called the competitive effect. We calculate the competitive effect as alij = aij/aii. The competitive effect of species j on i is the effect of j on i over the effect of i on itself. Essentially, αij represents the number of individuals of species i needed to equal the effect of one individual of species j on species i, and you can pretty much consider it the strength of interspecific competition. NOTE: that we are using a (alpha) instead of a, and i and j will be 1 or 2 depending on the species in which we're interested. 13. Using the data given above, calculate the competitive effects of species 1 on 2 and 2 on 1. LOTKA-VOLTERRA EQUATIONS Now let's look at the equilibria of two species simultaneously. To do this, we need to see how one population changes as the second population changes in response to the first, etc. First, let's fly through some math: Take our general competition equation from above: dN₁ N₁dt = - ra₁Na₁₂N₂ Now if we take that equation and divide both sides by a₁ we get: dN, =α11 N₁dt a11 T and K₁ = so a 11 a11 = K₁ And by similar logic: - N₁ α12 N2 = a11 = a11 011 Substituting gives: dN₁ _ r₁(K₁ – N₁ − α₁₂N₂) - N₁dt K dN₂ N2dt (K₂-N₂-α₂ N₁) K₂ - These two famous equations are generally called the Lotka-Volterra Equations, after the men famous for deriving them. These are just one model to explain competition between two species, and very often these equations do not do it well, especially when population sizes are far from equilibrium. But they are a great place to start. So let's figure out when the populations of both species are in equilibrium (i.e., no change in population size). 14. Assume both N>0, set both derivatives equal to 0 and solve for K for both species. (You only need to show the work for species 1.): You'll notice that your answers are linear relationships (as biologists generally write it, y = a + bx) where a is the slope. We can rearrange and solve for N and plot it. It's plotted for species 1 below.