Population dynamics Population size through time should be predictable N t+1 = N t + B + I - D - E Time 1 N = 100 20 births 25 deaths 10 immigrants 15 emmigrants Time 2 100 + 20 +10 25 15 = 90 Life History Tables However, birth rates, mortality rates, immigration and emmigration are variable by life stages Need to incorporate changing values to account for and predict age structure For simplicity, assume I=E Life History Tables Time (x) = time interval used for separating age categories. For simplicity assume t=1 (discrete generations). n x = number alive at age x l x = proportion of individuals alive at age x Age (x) nx lx 0 200 1.00 1 180 0.90 2 175 0.88 3 120 0.60 4 50 0.25 5 3 0.02 6 0 0.00 Life History Tables d x = proportion of original population dying during the age interval x to x+1 q x = proportion of existing population dying during age interval x to x+1; q x = d x /l x Age (x) nx lx dx qx 0 200 1.000 0.100 0.100 1 180 0.900 0.025 0.028 2 175 0.875 0.275 0.314 3 120 0.600 0.350 0.583 4 50 0.250 0.235 0.940 5 3 0.015 6 0 0.000 1
Life expectancy e x = T x / l x T x = average life expectancy from current time: e.g. how much living will be done by cohort from beginning of period x: T x =Σ(L x ); summed from x to last x L x =(l x +l x+1 )/2 Age (x) nx lx dx qx Lx Tx ex 0 200 1.000 0.100 0.100 0.950 3.130 3.130 1 180 0.900 0.025 0.028 0.888 2.180 2.422 2 175 0.875 0.275 0.314 0.738 1.293 1.477 3 120 0.600 0.350 0.583 0.425 0.555 0.925 4 50 0.250 0.240 0.960 0.130 0.130 0.520 5 2 0.010 6 0 0.000 Birth Rates and population growth f x = total natality; number of fertilized eggs produced in a given year by all individuals of age x m x = average natality of individuals of age x (f x /n x ) Reproductive Rate R 0 = rate of change in the population. If below 1.0, population is shrinking R 0 = (l x m x ) Sum of the number of fertilized eggs produced per original individual during each age Age (x) nx lx dx qx Lx Tx ex mx fx lxmx 0 200 1.000 0.100 0.100 0.950 3.130 3.130 1 180 0.900 0.025 0.028 0.888 2.180 2.422 2 360.00 1.80 2 175 0.875 0.275 0.314 0.738 1.293 1.477 3 525.00 2.63 3 120 0.600 0.350 0.583 0.425 0.555 0.925 4 480.00 2.40 4 50 0.250 0.240 0.960 0.130 0.130 0.520 5 250.00 1.25 5 2 0.010 6 0 0.000 R 0 = 8.05 Future population size N t = (N o * R o ) + I - E R o incorporates age-specific births and deaths Usually assume I = E for simplicity N t = (N o * R o ) N t = 100 R = 0.75 N 1 = 75 2
r and R o R o = net reproductive rate; for discrete generations (x=1) a multiplier allowing us to determine population size at future generation r (Malthusian Parameter) = intrinsic rate of increase; also per capita rate of increase. When r is >0.0 population will increase, when it is <0.0 population will decrease. r= ln R o /T Where T = generation time, time units between generations. For simplicity we assume this is 1.0 Intrinsic rate of population growth is defined as (Lotka- Volterra model): dn dt = rn or N t = 0 N e rt Sample calculations N t = (N o * R o ) N 1 =? N 0 = 100 R = 0.75 N 1 = 75 Assume T=1, then r = ln 0.75 / 1.0 r = -0.288 rt N t = N 0e N 4 = 100 e (-0.288*4) N 4 = 31.6 N 16 = 100 e (-0.288*16) N 16 = 0.99 Exponential Growth Human Population Growth 8000 6000 4000 2000 0 number of individuals10000 0 2 4 6 8 10 12 14 16 time r =.1 r =.2 r =.3 year r doubling time 1970 0.02 35 1991 0.018 39 2000 0.0125 55 Given current growth rates, what will the world population be in 30 years?? N t =N 0 e rt N t =6,426,101,450 e 0.0125(30) 9,349,922,439 3
Why don t we observe continuous exponential growth? Competition for limited resources Carrying capacity the number of individuals of a species that can be supported by available resources in a habitat Density dependent vs. density independent Both negatively impact populations growth/size If the impact worsens with greater density it s density dependent Disease Competition Famine If the impact does not vary with density it s density independent Disturbance fire, flood, etc. Density dependent effects Density independent effects Two natural populations showing exponential growth until K is approached. 4
Density dependent and independent factors A natural population showing density dependent effects. Incorporating Density dependent factors Lotka-Volterra Model dn dt = rn As you approach K, resources more limited, birth rates decrease, death rates increase. dn dt = rn 1 K N Intra vs. interspecific competition As N approaches K resources are more limiting, this is intraspecific competition Interspecific competition = competition among two species using the same resources Ecological equivalents: α 12 - Number of individuals of species 2 that are equivalent to one individual of species 1. α 21 - Number of individuals of species 1 that are equivalent to one individual of species 2. Types of Competition Types of resources Exploitative Use a resource more efficiently before a competitor has a chance Interference physically prevent a competitor from having access to a resource Asymmetric effect of species 1 on species 2 not the same as species 2 on species 1 Symmetric effects of species similar 5
Asymmetric competition -α 12 not equal to α 21 symmetric competition -α 12 roughly equal to α 21 Use α 12 to calculate affect of one species on another. K 1 =1000 N 1 = 600 N 2 = 300 α 12 = 0.8; 0.8 * 300 = 240 N 1 = 600 + 240 equivalent competitors = 840 Lotka-Volterra Models of Interspecific Competition dn dt 1 r N K N a N 1 1 12 2 = 1 1 K Models change in population size of species 1, accounting for impact of species 2. Similarly, affect of species 1 on species 2: dn dt 2 r N K N a N 2 2 21 1 = 2 2 K 1 2 Species abundance isoclines N 1 /K 1 =1 stable, all resources used by species 1 K 1 /α 12 =1 - stable, all resources used by species 2 (equivalent population) Combine the isoclines for both species to produce a graphical model of competitive interactions. Possible outcomes: -Stable coexistence -Dominance by one species 6
Competition and ecological gradients Models are oversimplifications, assume resources stable and consistent throughout Species are distributed across multiple gradients, should be most competitive (K maximized) near optima. Ecological Gradient Area with tolerable conditions Core area near optima Area with tolerable conditions Core habitat near optima Likely species distribution Second gradient 7
Niche combination of multiple optima along many gradients The role an organism plays in the environment All resources, interactions with biotic/abiotic components of the environment N-dimensional hypervolume Each dimension is a biotic or abiotic resource Niche Width Niche Width range of gradient(s) over which species occurs and is abundant. Generalist jack of all trades, wider range of optima, wider niche Specialist narrower range of optima, expect narrow niche Population Size Gradient Niche width and overlap along an ecological gradient Parameters d and w describe niche width and the amount of overlap among species. Non-competing specialists small w and large d (little or no overlap) Competing generalists large w and small d (large overlap) Niche space and competition Selection favors individuals who get the most resources Individuals that avoid competition will get more resources Competitive pressure leads to Niche shift Specialization 8
Evolutionary trade offs specialist vs. generalist Specialist (+/-) Competition in the intertidal zone What are some of the relevant ecological gradients in intertidal zones? What resources might be limiting? Generalist (+/-) Niche Shift through Character Displacement Character displacement selection for morphological change to relieve competitive pressure. 9
Fundamental vs. Realized Niche Fundamental niche total potential niche space for a species Realized niche actual niche space used, a subset of the fundamental niche. Convergent Evolution Similar niche properties exert similar selective pressure, resulting in similar species. Species no the same due to historical factors, continental isolation in this case. Predation Fundamentally, just another form of competition Involves energy transfer through consumption Carnivory Herbivory Parasitism Tertiary Consumer Predation and Natural Selection Predator selection for ability to obtain the most energetically beneficial food at the least expense. Select the most abundant, easiest to catch (old, young, sick, weak) Prey selection to avoid being eaten, or to become a less desirable meal. Secondary Consumer Primary Consumer Primary Production 10
Optimal Foraging Theory Predators should optimize energetic gains by balancing the costs/benefits of capturing prey. Costs Search time Handling time Digestion Benefits Calories assimilated 11