Population Growth: Exponential, Logistic, and Density Dependence, Slides of Ecology and Environment

Download Population Growth: Exponential, Logistic, and Density Dependence and more Slides Ecology and Environment in PDF only on Docsity! Unit II: Population Ecology • A. Single-species population growth • B. Predator-prey interactions • C. Interspecific competition • D. Mutualism Some population concepts • Population – a group of individuals of the same species occupying a defined location at a defined time; • Population density – number of individuals per unit area. • Metapopulation – a population of populations – a collection of spatially separated populations that interact with each other. Population processes • Population processes are inherently quantitative (count up numbers of individuals, of births, deaths, etc.); • These processes occur over large spatial and temporal scales; Population processes • Humans have difficulty observing phenomena on large scales; • An observer studying organisms cannot literally watch a population grow, even in bacteria • Models allow temporal and spatial scales to be compressed so that we can observe populations. In EVE 101, we will only “do” algebra but we need to understand calculus; Calculus is the field of mathematics that allows us to understand rates of change. Population Growth--review from 1/24/07 • Definition: Change in numbers of individuals through time, through births, deaths, immigration, emigration • A quantitative process-- “book-keeping” Simulation of DI population growth http://www.otherwise.com/population/exponent.html R = reproductive rate 1.x multipler, the net number of new individuals through time R = 1: 1 times any number is that number, so the number of fish stays the same. R > 1: the number of fishes increases: see how Play with different values of R to see what difference the scale of R makes in how fast populations grow. No matter how many fish there are, each individual gives birth to the same number of offspring The population grows faster and faster when R > 1 Density independent population growth and spread in space Time 3 Time 4 Time 5 Population starts here at time 1 at place X X Density independent population growth* and spread in space: another example: Homo sapiens *The recent trajectory of human population growth is density independent; the entire trajectory from early Homo sapiens to now is actually faster than exponential. Such a pattern is inverse density dependence. What limits population size? • Density independent limits: • Chance events • Per individual birth & death rates independent of density • What kind of environment? • Populations are periodically “wiped out” no matter what the population density. What limits population size? • Per individual rates dependent on density • As density increases: • Per individual birth rate decreases • Per individual death rate increases • EXAMPLES? Examples of density dependent population growth: sheep in Tasmania •An exponential (DI) trajectory at first •The population had a small “crash” after a high, but then leveled off fairly quickly This “sigmoidal” pattern suggests some kind of FEEDBACK between population density and vital rates (birth and death rates) Carrying capacity • Unlimited environments are unrealistic - populations cannot grow exponentially indefinitely. • Availability of resources (e.g. food, space) often limits population growth. • The carrying capacity (K) is the maximum population size that can be supported by available resources Mechanisms of density dependence (from The Science of Ecology, Ehrlich & Roughgarden) • Resource depletion (energy): – Effect on fecundity (reproduction) – Growth (future potential for reproduction) – Survival [all adds up to fitness!] • Space depletion • Increased time in social interactions • Waste product accumulation • Predation – Intraspecific (cannibalism) – Interspecific (predation, disease) • Dispersal (emigration) What kinds of things are going to apply to humans as population density increases? Outline to single species population growth topic I. Introduction: elements of population growth II. Types of population growth A. Exponential (geometric) -- Density independent PG B. Logistic -- Density dependent PG C. Examples of exponential and logistic population growth D. Mechanisms of density dependence III. Simple models of population growth A. Background B. Exponential C. Logistic PG and density dependence D. Allee effect (inverse density dependence) Explanation of model • “R” is formally called the net reproductive rate • R is called “net” because it is the net number of new individuals as some individuals are born and some die as time passes • R is a “rate” : • number of individuals per unit time N/T • R contains both births and deaths • R is (by convention) used for discrete time units, as in this model of geometric population growth: • Discrete time units increment in finite intervals, such as 1 year or 1 breeding season (usually are the same). Explanation of model • R ~ cB + p where – c is the survival rate of offspring; – B is the number of offspring per adult – p is the survival rate of adults • N(t+1) = R N(t) = (cB)N(t) + pN(t) in words: • The number of individuals present next year is determined by the number of surviving offspring produced this year plus the number of this year’s adults that survive until the start of next year. • This model fits synchronized generations such as short-lived herbaceous plants that live in a seasonal environment (annual, biennial, short lived perennial Ranunculus contains annual, biennial & perennial species. Simple DI model: Geometric population growth • N(t+1) = R . N(t) • N(t+1) =  . N(t) “lambda” is the traditional symbol for this model “t” is some meaningful interval, such as year (most organisms reproduce seasonally even in the tropics). • Solution over a long time, T, where t > 1 : N(T) =  T . N(0) after doing some math…. (Time passes: not just one time unit but many, perhaps equal to a generation or many generations) Density independence: “discrete” vs. continuous time • When Dt 0 (vanishingly small increments, I.e. “instants”) then we use calculus. Where r = instantaneous, per individual population growth rate “little r” = b - d (b is the per individual birth rate, d is the per individual death rate)  dN dt  rN  dN dtN  r r also called intrinsic rate of increase, Malthusian parameter, BIOTIC POTENTIAL Difficult to identify explicit generations in the population because generations overlap Continuous exponential growth dN/dt = r N Rate of change of population over time, t (change in numbers per time) Current population size (number of individuals) Per-individual rate of population increase Says that the current rate of change is proportional to the current number of individuals Continuous time = Population growth when individuals are not all synchronized If r = (b - d), when b = d, then r = 0 at exact replacement Integrate This is why it is called “exponential growth” Number of individuals at time t Number of individuals at initial time 0 Base of natural logarithm  dN dt  rN N(T) =  T . N(0) N(t) = N0 e rt Integrate over longer time periods Compare to geometric population growth: we’ll get back to this relationship soon! Relationship of “carrying capacity” and population growth • The ‘unused’ carrying capacity represents the potential for further population growth: K – N • This can be expressed as a proportion of the total capacity: (K – N) /K • This is termed the “environmental resistance” to further population growth Population growth in a limited environment The change in the size of the population (dN) over the time interval (dt) equals •the number of individuals (N) •multiplied by the rate of change per individual (r), biotic potential •multiplied by the potential for further population growth (K-N/K)  dN dt  rN K N K       Biotic potential Environmental resistance DD Population growth: play with algebra to understand how population growth changes with density  dN dt  rN K N K        dN dt  rN 1 N K        dN dt  rN  rN N K        dN dt  rN  r K      N 2 Direct density dependence: Effect of population density on PER INDIVIDUAL (CAPITA) vital rates (birth, death) Population growth rate ~ births minus deaths (b-d) r > 0 r < 0 DD Population growth: play with algebra to understand how population growth changes with density  dN dt  rmaxN 1 N K        dN dtN  rmax 1 N K       Population growth rate PER CAPITA (INDIVIDUAL) population growth rate  ractual  dN dtN  rmax 1 N K       rmax = biotic potential ACTUAL per capita population growth rate, given N, the population density Comparison of DI and DD population growth-- per individual ractual = N, population density rmax 0 Where is K? + - DI pop gr DD pop gr K  dN dtN b>d b<d Comparing population growth rates: per capita vs. overall population growth rate Population size (N) Population size (N) Population size (N) Population size (N) Population growth rate Logistic Growth K N = K/2 Population size (N) Population size (N) Population size (N) Population size (N) Exponential Growth Per capita growth rate K rmax  dN dt  dN dtN Population size N Inverse density dependence • Risks of small population size • What happens to dN/dtN when N is small?? • The “Allee effect” • = inverse density dependence • Named after the famous ecologist R.C. Allee who noticed that when populations are very small, “r” is depressed, r < rmax Inverse density dependence N ractual 0 - Slope positive K +  dN dtN When population sizes are low, we might expect the per capita growth rate to be less than the maximum. Daphnia example