Modelling Interspecific Competition

earlier this week we explained that populations can grow at geometric or exponential rates in the presence of abundant resources but as resources are depleted population growth slows and eventually stops as seen in models of logistic population growth while food may seem like an obvious example of a resource that individuals may compete for more generally a resource can be considered any feature of an environment that is required for growth survival or reproduction and can be used to the point of depletion recall this figure from our previous lecture that Illustrated the growth of a barnacle population the population began to level off after about 2 weeks but food was not the limiting resource space was eventually there wasn't any more space available on the rock for the larby to attach themselves this reduction in growth was the result of what we call intraspecific competition because it is competition occurring between individual of the same species this type of competition can cause density dependent reductions in population size such as those we explored in the last lecture now we are going to consider the effects on population size when individuals of different species are competing for the same resources which brings us to our first key concept interspecific competition occurs between individuals of different species that share the use of a resource that limits their growth survival or reproduction just as before this resource do does not have to be food or water we might have competition for space when that is in short supply as it is for these hard coral species or we might even consider competition for light among plants in the forest undergrowth any resource that can be consumed or depleted can provide a source of competition so I'd like to take a moment to introduce you to Robert MacArthur who even in his short life was a highly influential architect of modern ecology by helping to merge the roles of naturalists with those of theoretical biologists 1955 m Arthur was a graduate student at Yale University studying the Ecology of five warbler species that live together in the spruce forests of the northeastern us the Kate May warbler the black burnian black throated green the be breasted and the yellow rumped warbler now MacArthur noted that while density dependent events play a major role in regulating population size interspecific relationships were also important because the presence of an individual of another species may actually have some of the same effects as an individual of the same species now it was already accepted that two species using a limited resource in the same way cannot coexist indefinitely and this is a concept that we refer to as the competitive exclusion principle MacArthur questioned why competition wasn't excluding some of these species all of these species were about the same size and shape and they all feed on insects and to MacArthur it was puzzling that they could all live together in such an apparently simple habitat this he hypothesized that two or five in some cases species with identical ecological requirements would compete with each other and could not live in the same environment indefinitely and he decided to explore whether this hypothesis held true for these warbler species by categorizing their foraging Behavior MacArthur would take a tree and subdivide it into 16 zones based on height off of the ground and distance from the trunk height zones were measured in 10t units from the top of the tree and each branch was divided into zones that were the interior bare branches of the tree middle zones of old needles or The Terminal parts of the branch that contain the new needles he would then record the number of seconds that an individual spent in a particular portion of the tree this figure illustrates the data from one of those species the black throated green warbler let's consider for now just the left side of this figure which represents percentage of time and individuals spent in each of these zones the most concentrated activity was shaded until at least 50% of the activity was marked so you can see here that this species spent the majority of it time 40 to 50 met above the ground foraging primarily on branches with new needles these dat are simplified in this illustration where the peach color represents regions where the birds were found he repeated the same analysis for all five species and obtained the following results as you can see all of these species Focus their foraging in different zones extracting food from different parts of the forest if you will McArthur concluded that this resource partitioning reduced compet competion between the species and allowed them to coexist in the same location which leads us to our second key concept complete competitors cannot coexist competitors May coexist when they use resources differently now evidence for resource partitioning has been found in many other species including protests lizards fish Crustaceans and plants an overall study suggests that a species can coexist if they use resources in different ways an inference that is also supported by results from mathematical modeling so how can we model these types of interactions let's begin by revisiting our equation for logistic population growth and assume that in the absence of competition a species grows according to this equation then to model the effect of a competitor species the logistic equation can be altered in a way that mimics the effects of competition now in the late 1920s two researchers Alfred lka and veto voltera independently derived a way to use the logistic equation to model interspecific competition VTO voltera was actually an Italian mathematician and physicist in 1931 he was one of a small minority of Italian professors who refused to sign an oath of allegiance to musolini and was thus forced to resign his university position and go abroad he was stripped of all his Privileges and honors in Italian universities but later returned to Rome just before his death in 1940 Alfred lka on the other hand was not a scientist in the usual sense and wasn't associated with any University or scien ific institution he was actually a supervisor of the statistical Office of the Metropolitan Life Insurance Company of New York so he did have a very strong grasp of population statistics so let's begin by considering two separate equations to describe the logistic growth of two hypothetical species and throughout this presentation I'll be representing species one by Blue and species two Always by Green now the variables in these equations haven't changed K is still the carrying capacity R is still the intrinsic rate of increase and N is still the current population size however this format now acknowledges that the values may be different between the two species hence the one and the two so in other words this K sub one is the carrying capacity of species one which may be different than the carrying capacity of species 2 in order to account for the effects of competition we are going to add one additional variable to each of these equations and we refer to this variable is the competition coefficient for species one this coefficient is represented by Alpha which is the effect an individual of species 2 has on the population growth of species one now note in this equation that we are multiplying Alpha by N2 or this population size of species 2 this makes sense because you're considering the effect that one individual of species 2 has Alpha and multiplying it by the number of individuals in species 2 and two and the product will then be the cumulative effect of the entire species 2 population on species one so we can do the exact same thing to the species 2 equation but we will use the coefficient beta and then multiply that by N1 now if this seems a little confusing or a little abstract I completely understand so let me just present a quick example that might help put things into perspective so let's imagine that we want to model the competition between a population of gray squirrels which will designate a species one and chipmunks that will designate species 2 so here's our equation for species 1 that we were using before and recall that Alpha is the competitive effect that a single chipmunk from species 2 has on the squirel population so this measures basically the intraspecific competition relative to the intraspecific competition so in other words how many individuals of species 2 or Chipmunks are equivalent to one squirrel in terms of their use of the resource so in terms of this common resource use one chipmunk might be hypothetically the equivalent of a quarter of a squirrel so it would take four Chipmunks competing with a squirrel to equal the competition from another squirrel thus the coefficient would be 0.25 we can then take that Alpha coefficient and multiply it by the number of chipmunks to get the overall effect in Squirrel equivalent that the Chipmunks have on the squirrels these equations can provide us with a very powerful tool to predict the outcome of competition so if we know how each species population is changing and the competitive effect they have on each other we can then determine whether one species is going to win the competition and drive the other to Extinction or if they will both be able to coexist so we'll Begin by determining when the population of each of these species would stop changing as we saw in the logistic growth model D n/ DT represents the rate of change in population size at each instant in time when dn/ DT equals z the population size stops changing and remains constant unless perturbed in some way so for example if many individuals died in a catastrophic fire or a storm right now this equation is just a simple product of three values and as I'm sure you remember from basic math substituting zero for any one of these would make this statement true now we have two options the first setting N1 equal to zero simply States the obvious a population with no members does not grow the second option would be to calculate what would make this red highlighted part of the equation equal to zero now obviously we can't have a carrying capacity equal to zero or this fraction would be undefined so we have to set the numerator equal to zero and solve for N1 this gives us what we referred to as the zero population growth isocline for species 1 when the species one population is equal to its caring capacity minus the product of the competitive effect of species 2 it will be stable in population and not change now logically this also makes sense when we were just considering the logistic growth model the stable population size was the carrying capacity now that we have competitors to worry about the stable population size is being reduced we can do the exact same calculation for species 2 to check your understanding pause this video and go back to calculate the zero growth isine for species 2 resume the video when your calculation is complete hopefully you were able to derive an isocline of K2 minus beta * N1 so recall from the last lecture that under the logistic growth model alone if individuals were only competing with members of the same species the population would be stable if it was equal to the carrying capacity so in this case N1 would equal K1 or N2 would equal K2 now when interspecific competitors are in the picture the population needs to be less than the carrying capacity in order to be stable the environment can't sustain a population as large as the carrying capacity due to the presence of an interspecific competitor find it helpful to think of this as a modified or a realized carrying capacity if we increase N2 so we're adding more species to or adding more competitors the stable population of N1 will decrease and vice versa now you might have noticed that both of these are simply equations for lines Alpha and beta are constants as are the carrying capacities but we can think of N1 and N2 as values on an XY AIS in order to plot these lines so let's begin by just considering the isine for species one note that this line intersects the horizontal axis when N2 equals 0 or the population size of species 2 is z we can plug zero in for N2 in our equation and then solve for N1 to get that intercept and as you can see that value is k1 or the carrying capacity of species one this probably isn't surprising to you if there were no individuals of species 2 there would be no competition and species one would be able to reach its caring capacity and stabilize at that level what about the other intercept why don't you pause this video and see if you can solve this on your own I'm sure that wasn't too much trouble hopefully you plugged in zero for the N1 population size and proceeded to solve for N2 arriving at K1 over Alpha we can complete the exact same analysis with the species to isocline only now the species of concern is on the vertical axis try to solve for these intercepts on your own and resume the video when you've completed this hopefully you were able to determine that when N1 equals z the isocline intercepts the N2 axis at a value of K2 or the carrying capacity of species 2 when N2 equals z the isocline intersects the N1 axis at K2 over beta to keep things simple for now we are just going to consider a single isocline and I'm going to remove the intercept labels here's the line represented by the species one isocline as a function of the population sizes of species one and the population sizes of species 2 the population of species 1 will be stable and unchanging at any combination of N1 and N2 that falls on this line so hypothetically if species one has this many individuals the population size will be stable and unchanging if species 2 has this many individuals or in other words while the population might otherwise be undergoing logistic growth in the absence of species 2 the effect of competition from species 2 prevents that growth from occurring and keeps the size of species one stable what if the combination of N1 and N2 don't lie on the line by definition because this is an isocline of zero growth that means that N1 is not stable and will change if this point is not on the line but will it increase or will it decrease let's keep the size of species one the same but let's reduce the size of speci species 2 now our point would fall below this isine we just said that if the point is not on the isine the species one population will change because this point is located below the isocline the population size of species one or N1 is going to increase this is a generalization that will hold true whenever you are examining a point relative to an isocline if a species population size is below its own isocline the population size will increase now this makes both logical and mathematical sense if the number of competitors was reduced below the level required to keep species one in check species one will be able to grow in size so revisit our modified equation for population growth that you saw a few slides ago now recall that under zero growth conditions this numerator was equal to zero if we reduce N2 the numerator of this fraction will go from being zero to a positive number which will in turn change DN over DT from zero to a positive number the population size of species one will now get larger the exact opposite is true if the combination of populations places the point above the isocline so let's assume that we have an N1 the size of this in order to fit the zero growth isocline species 2 would have to have a population size right about here but what if there were more of species 2 than that now we have a point that is above the species 1 isocline again if the point is not on the species one zero growth isocline the species one population will change because this point is located above the isocline the population size of species 1 or N1 is going to decrease this is a generalization that will also hold true if a species population size is above its own isocline the population size will decrease notice that our increase and decrease arrows are only pointing left and right this is because right now we are only considering the species one isocline and what happens to N1 because this population is plotted on the horizontal axis moving to the right is increasing and towards the left decreasing I think you can probably guess where we are headed next from there we can complete the exact same analysis with the species 2 isocline only now the species of concern is on the vertical axis if the N1 N2 population plot point is below the species 2 isocline species 2 will increase if it is above species 2 will decrease so in order to make predictions about a population's trajectory we need to know a what the current population size is of species one and species 2 and B whether that point is above or below each species isocline when we plot both isoclines together they can take one of four orientations in the first option the isoclines do not cross and the species one isocline is above the species 2 isocline in this orientation species 1 will always win and drive species 2 to Extinction so how does this occur let's first consider what happens if the populations of species one and two were to be plotted somewhere in this Zone note that we are below both isoclines so species one is going to increase and species 2 is going to increase so our next Point might be somewhere about here in this blue shaded region we are above both isoclines so species 1 will decrease species 2 will decrease and our next Point might be here in this orange region it is below the species 1 isocline so species 1 will increase but it is above the species 2 isocline species 2 will decrease this trajectory will continue until species 2 is eliminated at which point species one will be able to stabilize at its carrying capacity note that this option does not require the two iines to be parallel only that they just cannot cross the second option is similar to the first in that the isoclines do not cross but in this case the species 2 isocline is above species one the outcomes in the blue and green regions are the same as before because we are still either above or below both isoclines however this time in the middle we are below isocline 2 and above isocline 1 so the the exact opposite happens species 1 decreases and is eventually driven to Extinction as species 2 increases and stabilizes at its caring capacity species 2 wins I think you're starting to get the hang of this now so time to get a little tricky what happens if the isoclines cross note that in this option the carrying capacities are the highest intercept on either axis now we have four regions to consider using the same method of analysis as options one and two consider the direction of population change in each of these regions pause the video and take a moment to sketch these arrows on a sheet of paper the blue and green regions are pretty straightforward as before however the overall winner is determined by which of the two orange regions the population trajectories enter first if it's the dark orange we are below the species 1 isocline above the species 2 Bine so species one will win the exact opposite will happen with the light orange section in other words one species will win which one depends on how their populations change over time and finally option four the isoclines are still crossing but this time the carrying capacities are the lower intercepts in this case the population trajectories will lead both species populations to the intersection of the isolines once here because we are on both is lines neither population will grow any further and they will stabilize at this equilibrium density note however that the final density of each is lower than the carrying capacity of either this is the only scenario in which competitive exclusion does not occur time to check your understanding with two different examples these problems can also be found in a separate file on blackboard in the Land Between the Lakes elk and bison Prairie we currently have 125 bison and 15 elk these species are known to compete for some but not all of their resources we are able to determine that the carrying capacity of the Bison K1 is 200 and that the alpha competition coefficient is four the carrying capacity for Elk K2 is 100 and the beta competition coefficient is one over time will both species be able to coexist if not which one will be driven to Extinction in this area so how do you approach this type of problem you saw early in this presentation that we can determine the isocline intercepts if we know the carrying capacities and the competition coefficients Begin by using the given information to plot your isolines on a graph the values you calculate for the intercepts will help you determine whether they Cross or not and which one is above the other now consider your starting population sizes N1 the Bison is 125 and N2 the elk is 15 where does this plot on your graph in other words is it in the blue the green or one of the orange regions what will be the outcome of this competition question two a small lake is capable of supporting both bluegill and smallmouth bass these two bony fish species compete for similar food resources the carrying capacity of this Lake for Bluegill K1 is500 and for bass K2 is 500 after some careful measurement we find that Alpha is 0.5 and beta is 0.25 we stock the lake with 250 bluegill and 750 bass over time will both species be able to coexist if not which one will be driven to Extinction in this Lake if they are able to coexist approximately how many of each species will be present the answers to both of these questions are due in class on Thursday please include a sketch of your isocline graphs along with any math to support your conclusions thanks for watching and see you all soon