David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems• Recall our definition of ecosystems:–The minimal systems on Earth that exhibit a flow of energy and a complete chemical cycling are composed of at least several interacting populations and their non-biological environment• One of the key ways in which we find populations of biological organisms interacting is by predation:– Living things consume one another for survival; One creature captures and devours (the predator), another creature is the one captured and devoured (the prey)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems•In stable ecosystems, we find the populations of predators and prey in a symbiotic balance, where the size of one population is regulated by the size of the other• In terms of system behaviors, this was the example we used for an oscillating system:David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems• One of the other themes we discussed earlier was human modification of ecosystems:• We can and have interfered with many predator-prey relationships, such that the system is pushed far enough away from the symbiotic balance that it cannot recover, and other behavior results– E.g. we can destroy the habitat of the prey, leading to the population of the prey perishing, with the predators quickly following, OR– We can remove predators from the system, leading to the prey population moving entirely out of control– And, the Principle of Environmental Unity tells us that these consequences can in turn lead to further changes …David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• For our examination of predator-prey systems, we will use a theoretical example:• Verdant National Park has a problem with its deer population:– Historical records suggest there were wolves in the vicinity, but that population was removed a long time ago, such that there are no natural predators of deer– The deer population has been growing progressively larger– Vegetation has been overgrazed to the point of certain species are in real danger of disappearing from the park– Car accidents between park visitors and deer are becoming extremely common• Strategy: Control deer by reintroducing wolvesDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• Before we can get a sense of how the integrated predator-prey system functions, its useful for us to first look at how an individual population’s dynamics function in a system that is in balance– E.g. in an ecosystem without a food shortage and some predation, what would a prey population’s dynamics look like?David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• We can write the difference equation that expresses the size of Population P at a future time (t+∆t):P(t+∆t) = P(t) + B * ∆t – D * ∆twhere P = the populationB = the number of births per time unitD = the number of deaths per time unit• Quite often, we find the number of births and deaths per time unit are a function of population density, and assuming the region we are considering is of a fixed size, these are a function of population size– Explanation: Higher population densities lead to more mating opportunities (birth), and also more intraspecies competition (death)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• We can describe both of these situations as linear relationships between the process and population:B(t) = bP(t)• I.e. Births per unit time expressed as a linear functionof the population size, according to birth rate bP(t)B(t)B(t) = bP(t)• Similarly for death per unit time:D(t) = dP(t)• I.e. Deaths per unit time governed by death rate dP(t)D(t)D(t) = dP(t)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• In this example, we can describe the relationships between birth/death and the population size in terms of linear relationships, but we can (and later will) represent these relationships using other functions• When we represent these relationships as linear, we have defined a system that will exhibit exponential growth behavior (assuming birth rate > death rate), and can use the difference equation from earlier, substituting in the linear functions:P(t+∆t) = P(t) + B * ∆t – D * ∆tP(t+∆t) = P(t) + b * P(t) * ∆t – d * P(t) * ∆tP(t+∆t) = P(t) + (b – d) * P(t) * ∆tDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example• We can get from the difference equation to the rate equation using the usual method:P(t+∆t) = P(t) + (b – d) * P(t) * ∆tP(t+∆t) - P(t) = (b – d) * P(t) * ∆tP(t+∆t) - P(t) = (b – d) * P(t)∆tP(t+∆t) - P(t) = (b – d) * P(t)∆tLim∆t Æ 0= (b – d) PdPdt• The rate equation shows that the change is proportional to the population size P, and the rate constant (b – d)David Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Unconstrained Example•We can interpret the rate constant such that we can see what the fate of the system will be:– If (b – d) > 0 then the right side of the rate equation will be positive, and the population will experience continuing growth– If (b – d) < 0, then the right side of the rate equation will be negative, and the population will decrease and will eventually go extinct•We can solve the rate equation to yield an expression for the size of the population in terms of the initial population size, the birth and death rates, and the time elapsed:P(t) = P0e(b – d)tDavid Tenenbaum – GEOG 110 – UNC-CH Fall 2005Modeling Predator-Prey Systems -Constraints on the System• Theoretically, populations are capable of exponential growth, but this is rarely achieved in nature for long• In practice once the population gets to some size, we see some control of the population• E.g. once the deer population gets large enough, overgrazing occurs, and there simply isn’t enough food available to support that number of deer Æ We observe intraspecies competition at high enough population densities, along with higher incidence of contagious disease because of crowding• We can express this limitation in terms of carrying capacity, as we did when describing