Density-Dependent Population Growth All populations are constrained to a maximum size based on available resources (e.g., food, water, thermal and safety cover). We call this size, beyond which a population cannot sustain itself for any extended period, ecological carrying capacity (K). It is important to understand that carrying capacity is a feature of the species' habitat. Populations cannot grow indefinitely and can only be briefly be over the ecological carrying capacity, if at all. In general, populations - absent some unusual force such as extensive human harvest, catastrophic weather event, or introduced predation or disease - should be expected to increase up to near their carrying capacity and then maintain themselves at or near carrying capacity. We do not expect to see populations above or well below carrying capacity in the absence of some extenuating circumstance. Density-independent (geometric or exponential) growth predicts an ever-growing population when r is greater than 0 (or > 1). This may be useful for managers as a model of growth for very short periods and conditions, but we would prefer to have a model that stabilizes a population near K, as we observe populations doing in nature. Density-dependent models do this, and provide useful inferences towards management of wildlife populations. Basic model of population growth has the following assumptions: 1) The population is geographically closed (no immigration or emigration) 2) Birth and death rates are constant a. across individuals (no variation among ages, sexes, etc) b. across time c. across space d. across population size Based on these assumptions, the basic BIDE model becomes Nt+1 = Nt + B - D. If we want a population to stabilize at or near its carrying capacity, it should be apparent that at K birth rate must equal death rate, or b=d. In that case the above model becomes Nt+1 = Nt, the population is stable. For this to occur, we need to relax assumption 2d above - birth and/or death rates need to vary -as population size increases either birth rate has to decline, death rate increase, or a combination of both. This is known as density-dependence (specifically, negative density dependence because overall growth rate declines as the population becomes larger. We model this process using the density-dependent growth model, commonly referred to as the 'logistic growth model'. The most common formulation of this model is: ???????? ???????? = ???????????????? (1 ???? ????). Those that have had calculus might recognize the ???????? ???????? as a derivative. This basically means the 'change in population size over an extremely small period of time'. K is the carrying capacity and N is the current population size. The rmax is the maximum growth rate a population can show under perfect conditions, called 'biotic potential' and sometimes noted as r0. This form of the equation works well to show how the logistic model works - we start at very small population sizes with growth rates at their maximum. As the population grows, the (1 ???? ????) part of the equation gradually reduces the observed growth rate of the population. Sometimes this part of the equation is referred to as 'environmental resistance' as it is incorporating 'how hard' the environment is reducing the maximum possible growth rate. For example, if a given environment has a carrying capacity of 500 for a given species, then at population size 0, we get (1 0 500) = 1. At very low population sizes we are multiplying rmax by a number very close to 1, so we expect population growth near rmax. At carrying capacity we have (1 500 500) = 0 so the observed growth rate is zero times rmax, or zero - we see no population growth. At a population size of one-half carrying capacity we get (1 250 500) = 0.5 so we see a growth rate of one half of rmax. There are a few things to keep in mind about this equation, focusing on the right-hand side, ???????????????? (1 ???? ????). Perhaps the most important is to keep the observed growth rate, rrealized, clearly differentiated in your mind than rmax - they are not the same thing! The entire right-hand side of the equation represents rrealized or ???????????????????????????????????? = ???????????????? (1 ???? ????). Second, population growth stabilizes when ???????????????????????????????????? = ???? ???? = 0. Observed population growth is the combined result of births and deaths (and immigration and emigration). If we graphed birth and death rates over a range of population sizes, the birth rate line will be above death rate when r>0, will be below death rate when r1 when below carrying capacity, and 1 density-dependence has little effect on population growth rate until it begins approaching carrying capacity, a form known as convex or weak density dependence, while when