Exponential Population Growth: Understanding Exponential Growth Models and Consequences, Exercises of History

Download Exponential Population Growth: Understanding Exponential Growth Models and Consequences and more Exercises History in PDF only on Docsity! 1 Lecture 14: Population growth. Outline Exponential growth described by R0, λ and r Ro - Discrete breeding seasons, nonoverlapping generations, semelparous life history λ - Discrete breeding seasons, overlapping generations, iteroparous life history r - Continuous breeding seasons, overlapping generations, iteroparous life history General properties of exponential growth models Consequences of exponential growth Density-independent and density-dependent limiting factors Density dependent population growth: logistic equation Continuous breeding seasons, linear density dependence (Verhulst-Pearl eqn) Discrete breeding seasons, linear density dependence Discrete breeding seasons, nonlinear density dependence Compensation, overcompensation, undercompensation Limitations, assumptions, usefulness of these models Introduction The primary goal of this section is to understand the processes that cause changes in population size through time. A second goal is to understand simple mathematical models that describe population growth. We’ll start with the simplest models for single species, and add to them to become progressively more realistic (adding effects of intraspecific competition, interspecific competition & predation). A lot of the theory underlying ecology and population biology is based on extensions of these models, so it is critical to understand them well. Exponential population growth (Pianka Fig. 9.1) The simplest type of population growth is exponential, as shown by reindeer on Pribilof Islands for 30 years after introduction. Exponential growth occurs when a single species is not limited by other species (no predation, parasitism, competitors), resources are not limited, and the environment is constant. These conditions called an ‘ecological vacuum’, and this does not often occur (for long) in nature. But colonizations like the reindeer example, or recovery of a population after a large-scale disturbance (fires, crash from disease) can allow exponential growth for a period. 2 How do we build a model of exponential population growth? 1. Consider an animal like Antechinus, Australian marsupial mouse. Antechinus are seasonal breeders that mature at age 1 year. Males mature, fight like mad, mate like mad, and all die (due to a huge pulse of the stress hormone corticosterone). Females live a few weeks longer, but just long enough to raise a litter, then die after weaning. This life history shows: - discrete breeding seasons - nonoverlapping generations - semelparous life history How do you know if an Antechinus population is growing or not? From demography lectures, R0 =  lxmx. R0 is the average number of offspring produced by an individual in its lifetime, called the net reproductive rate, or the net replacement rate. As long as survivorship (lx) and fecundity (mx) stay the same, R0 can be used to project the number of mice in the population one generation from now: N0 = number in population now N1 = number in population one generation later N1 = N0R0 In turn, the population of N1 will grow by the same rule (initial population size * R0) over the next generation: N2 = N1R0 Substituting for N1 gives: N2 = N0R0 2 Generalizing: Nt = N0R0 t Which gives the population size t generations in the future. This is the most basic discrete population growth model, from which all others are derived. It models exponential growth because it assumes that R0 is constant  it assumes that survival and fecundity do not decrease (or increase, for that matter) as the population gets larger. 2. The marsupial mouse has unusual life-history for a mammal (though it’s common among invertebrates). A more typical mammalian life history is shown by African wild dogs, which breed one a year (in the dry season, when prey is more easily caught), but individuals breed repeatedly in a lifetime, and have overlapping generations. 5 Consequences of exponential population growth. How long does an exponentially growing population take to double? This is equivalent to asking how long t is, when Nt = 2N0. Using exponential growth equation, Nt = N0e rt , substitute 2N0 for Nt, 2N0 = N0e rt ln2 = rt t = ln2/r doubling time = 0.7/r This equation for doubling time works for any process of exponential growth. Some everyday examples: 3.5% inflation means that prices will double (value of dollar will halve) in 0.7/0.035 = 20 years. Typical credit card debt has annual interest (APR) of 18%. The amount you owe doubles in .07/0.18  4 years (!) Human energy use has been increasing at 5% per year. In the next 0.7/0.05 = 14 years, energy use will double. Atmospheric CO2 is increasing at about 1% per year, so the atmospheric CO2 concentration will double in about 0.7/0.01 = 70 years. 2. Point 1 makes it clear that exponential quickly growth accelerates to become extremely rapid. Because of this, exponential growth is always temporary, and depends on the existence of an ‘ecological vacuum’. As a population grows, it will eventually be limited by one or more ecological factors (e.g. shortage of food). Density-dependent and density-independent limits on population growth What stops exponential growth, or prevents it from beginning at all? Remember that exponential growth models assume that birth and death rates are constant (r or R constant). In the real world, birth and death rates change over time, and these changes can limit population growth. Two general classes of limiting factors: 6 Density-independent limiting factors: reduce population growth regardless of population size. Density-independent limiting factors: 1. Are usually physical in nature (hard winters, failure of rainy season). 2. Are more important for small organisms, because small organisms are not as well buffered against physical environment. 3. Are more important in extreme or highly seasonal environments than in mild, stable environments. 4. Can interrupt exponential growth or cause declines, but cannot regulate a population at a stable population size. Density-dependent limiting factors: reduce population growth with an impact that depends on current population size. Examples: (Fig 2-11 Gotelli) survival and reproduction decrease as population size increases in Song Sparrows on Mandarte Island (Tables 17-2 and 17-3 Ricklefs) reproduction in w-t deer declines as population density increases (Fig 17-7 Ricklefs) experimental example: mortality increases as density increases in grain beetles. For all of these examples, the density-dependent limiting factor is probably intraspecific competition for limited food. Density-dependent limiting factors: 1. Are usually biological in nature (competition, disease, predation). 2. Are more important for large organisms (which are buffered from physical environment). 3. Are more important in physically benign and constant environments. 4. Can interrupt exponential growth or cause declines, and CAN regulate a population near a stable population size. Density-dependent population growth. A (closed) population is growing when births exceed deaths: r = b - d > 0. Population Size, N Time, t 7 If birth or death rates are affected by density dependent factors, plotting b and d against population size predicts where per-capita birth and death rates will exactly balance and population size will stabilize (dN/dt = 0). The stable point is called the carrying capacity, K. Per capita birth rate, b death rate, d Population Size, N Viewing this in terms of number of individuals, rather than per-capita rates, see that density dependent growth follows an S-shaped “logistic”, “sigmoid” curve. Logistic growth curves are common in nature. Logistic population growth models How do we modify exponential growth models to take account of density-dependent limits on growth? 1. This is easiest to show for continuous breeding seasons. The exponential growth model for continuous breeding is: dN/dt = rN Simplest way to incorporate density dependence is to assume that b declines and/or d increases in straight line fashion as N increases. (see two figs previous) In this case (linear density dependence), actual per-capita rate of increase  r when N = 1 K dmin bmax 1. birth rate b declines as N increases 2. death rate d increases an N increases 3. when b=d, growth rate r =0: this is the carrying capacity, K N t K dN/dt/N or realized growth rate, (ra) N N= K dN/dt/N = rmax