Discrete Population ModelsA population is a group of individuals that belong to a single species that live in some defined area.Let the size of the population at some specific time, t, be NtThe size of the population at time, t + 1 (this could be 1 year, 1 month, or 1 day after time t; same as the unit for t), is Nt+1What controls Nt+1? What can cause a population to change?Nt+1 = Nt + births – deaths + immigrants – emigrantsFor our purposes, we will consider a closed population (we will ignore immigration and emigration); this just leaves births and deaths.Question – do you think the number of births or deaths is independent or dependent on population size?It’s dependent – for example, you can’t have more deaths than members of the populationLet  = constant birth rate (births per individual per unit time ) and  = constant death rate (deaths per individual per unit time). Then in the interval of time ]1,[ tt, tN births and tN deaths occur.Hence, Nt+1 = Nt + tN )(We can simplify this to:Nt+1 = λNt (called a recurrence relation)where  is a constant.  is rate of growth per individual if 0, and  is the rate of decay per individual if 0.What would the population be after two unit time intervals? Nt+2 = λNt+1 But what is Nt+1?Nt+2 = λ(λNt) = λ2NtThis sort of approach is known as recursion. A recursive equation is one which you keep repeating over and over again until you get to the final answer. If I asked you what Nt+187 was, you could figure it out by just repeating the formula 187 times. Computers could calculate this easily. What about a third time interval?Nt+3 = λNt+2 = λ(λ2Nt) = λ3NtDoes anyone see a pattern?Nt+T = λTNtFor simplicity sake, we will assume that the starting time is always t = 0, thereforeNT = λTN0This is all based on discrete time intervals; it can also be done using continuous time intervals, but that involves calculus which we will not go into hereWhat does this look like? The following is the shape of a plot when N0 = 10 and λ = 2.The horizontal axis is the time-axis and the vertical axis is the population-axis.Some advantages of logarithmic axes.If we plot the population size on a logarithmic axis, we find the followingFirst, the growth curves is now linear. Second, if N0 is the same, but λ varies, the lines have the same intercept, but different slopes (the slope is equal to log λ).If λ is constant, but N0 varies, the lines are parallel. N0 is equal to the y intercept.What is the equation of a line? Y = mx + b (m = slope, b = intercept)Applying logs to the equation NT = λTN0 , we get Log NT = Log λTN0 = Log λT + Log N0 = T Log λ + Log N0Sample problem:One hundred rabbits are introduced onto a small island at the beginning of 1995. If this population grows with a geometric growth rate of λ = 1.3, how many rabbits will there beat the beginning of 2000? What about the beginning of 2005?We use the equation 0NNTT to solve the problem. T = 0 represents the beginning of 1995. So, T = 5 represents the beginning of 2000. Hence, we have 055)3.1( NN  = (3.713) × 100 = 371 rabbits (in 2000)01010)3.1( NN  = (13.786) × 100 = 1379 rabbits (in 2005)Question: How long did it take the rabbit population to double in size? i.e. for what valueof T does NT = 2N0? .NT = λTN0 = 2N0λT = 2T ln λ = ln 2T = ln 2 / ln λ = 2.6 yearsHow long will it take the population to increase 100-fold? Using the same logic, we’d find T = ln 100 / ln λ. If λ = 1.3, T = 17.6 yearsExercise:Suppose that the number of bacteria in a certain colony quadruples every hour.a) Set up a recurrence relation for the amount of bacteria after t hours have elapsed.b) If 200 bacteria are used to begin a new colony, how many bacteria will be in the colony in 8 hoursWhen is the population in equilibrium? Remember, equilibrium means the population is not changing through time: NT = N0This occurs when λ = 1 or when N0 = 0 (i.e., the population is extinct). Verify this.A plot of Nt+1 v/s Nt gives us a graphical way to show population growth.The basic diagonal indicates when Nt+1 = Nt. The other line has a slope equal to λ. To show how the population changes, we find N0, move up to the λ line, then move over to the equal line. This gives us the population at N1, we can keep repeating this to show how the population changes over time.What happens when λ is less than one? The population decreases.Note what happens if λ = 1. (Equilibrium)Everything up to this point has assumed that the population at some time t+1 is dependent only on the population at the previous time t, i.e.Nt+1 = λNtWe’re going back to uniform populations and ignoring age and everything again. If λ equals 1 the population is in equilibrium; if λ is less than 1 the population will decrease to0. What happens if λ is greater than 1? The population will increase to infinity. Is this realistic?Various forces act to keep a population size down. The implication is that λ is not constant. The rate of growth is itself dependent (a function) on population size;λ = f(Nt). In the simple case it was λ = 1 + R., R was birth rate – death rate so λ = 1 + birthrate – deathrate.For λ = f(Nt) to be true, either birthrate or deathrate (or both) must be dependent on population size. Let’s think about it. Why might the death rate change when the population size goes up?Less food, water, etc. More animals starve. More fighting for resources. Why might the birth rate change when the population size goes up?Poor resources means fewer offspring; offspring are more likely to die. Maybe there are limited breeding spots (e.g., nest sites in birds) and so a smaller percent of the population is actually capable of breeding.This is all known as density dependence. That is, the rate of change is dependent on the density of the population. Define K as the carrying capacity…roughly the maximum number of individuals that can be supported by the environment. 11 1 1t tt t tK N NN N R N RK K+� - � � �� � � �= + = + -� � � �� � � �� � � �� � � �Note that λ(Nt)= 1 1tNRK� �+ -� �� �. Before it equaled 1 + R.When Nt is small, the equation is approximately 1 + R (the rate of growth is unlimited). When Nt is large, the rate of growth becomes 1 (the population is in equilibrium).Unlike before, there is no simple solution to this equation for an arbitrary number