Question

These relate to the Lotka-Volterra equations we discussed for competition and for predation. You may have to use web resources for some questions.

For two competing species, use what you know about Lotka-Volterra models to sketch and describe the outcomes for each of the following scenarios:

What would be the ultimate outcome for both competitors in each of the following sets of conditions? (15 points total).

12. K1 = 200, K2 = 100, alpha = 1.5, beta = 2

Species 1 will win the competition.

13. K1 = 100, K2 = 200, alpha = 1.5, beta = 2

Species 2 will win the competition.

14. K1 = 200, K2 = 100, alpha = 2.5, beta = 1.5

Both species have greater competition with each other, so there is UNSTABLE coexistence.

Remember, I used β (beta) but your book uses alpha (α1,2 and α2,1). It’s just two ways of saying the same thing.

Let’s take it up a notch. Consider the following scenario: For many years, the population size of Species 1 has consistently hovered around 100 individuals/km2 while Species 2’s population size has been approximately 50 individuals. Species 1 uses resources at one-quarter the rate of Species 2. Use the information to answer the following questions. You’ll need your critical thinking skills!

Given the information available, sketch the Lotka-Volterra isocline diagram that best represents the scenario above. Don’t forget to label your graph. Are the populations of each species likely to be stable or unstable? (5 points)

Now imagine a case where the values above are the carrying capacities, not the population sizes. For simplicity’s sake, assume that resource use is the only factor that determines competition coefficients (alpha and beta). Sketch the isocline diagram that describes this new scenario. Which species should win? (5 points)

BONUS (3 points) What would be the population sizes of each species in the absence of their competitor? In other words, if I removed individuals of Species 1, how many individuals of Species 2 could there be? Likewise, if I removed individuals of Species 2, How would Species 1’s population grow? (Hint: it might be helpful to first consider the effect of removing a single individual).

Explain how the two figures below relate to one another in terms of predator-prey relationships? Label the axis and put in any missing directional arrows (5 points)

Answer 1

Lotka-voltrra model of interspecific competition is a  simple mathematical model which represent the interaction between 2 competing species .It is based on  verhulst-pearl equation----

dN/dT=rN (K-N)/K.

In the logistic equation , population growth rate is limited by intraspecific competition.Lotka-volterra model was developed to allow ecologist to predict the potential outcome when 2 species are in competition for the same resources.

1-The logistic equation for species 1---dN1/dt=r1N1 (K1-N1/K1)

The Lotka-volterra equation for species 1 in presence of species 2 ---

dN1/dT=r1N1 [K1-N1-12N2/K1]

2 --The logistic equation for species 2 growing alone dN2/dt=r2N2 (K2-N2/K2)

The Lotka-volterra equation for species 2 in the presence of species 1--

dN2/dt=r2N2 [K2-N2-21N1/K2]

Here-- N1&N2 = population of species 1 and 2.

K1 &K2 = carrying capacity of system for each species in the absence of other

t= time

r1 & r2 = per capital growth rate of species 1 and species 2

Alpha 12= is a competition coefficient representing the effect of one individual of species 2 on the growth of species 1.

Alpha21=is competition coefficient representing the effect of one individual of species 1 on the growth of species 2