BS3120 - Species Modelling

BS3120 - Species Modelling

1. Growth

   • Exponential Growth
     • Rate of growth is proportional to population size; larger population = greater rate of growth. Plotted against logarithm = straight line w/ gradient r. Accelerated growth as long as resources aren't limiting. Rare but does happen (Eg: Insect swarms, weed growth, human population).
     • dN/dt = rN
       • Change in population size in a given time frame = growth rate x population size.
   • Logistic Growth
     • Population growth continues until size is too great and then growth slows the equilibrium.
     • Density (In)Dependence
       • (In)Dependence in relation to the size of the population Independent = Life history Dependent = Resources, mates, birth/death rate potentially.
     • Carrying Capacity
       • = K. Maximum population size which can be sustained with the provided resources.
       • K = (b-d)/(a+c)
         • b-d (birth - death) = growth rate. a + c = strength of density dependence. More density dependence = lower carrying capacity.
       • Chaos
         • Ratio between time lag and response time (growth rate) determines growth. Small = smooth rise, settle at K. Medium = first overshoot, then undershoot. High = Stable limit cycle, k never reached and is the midpoint between oscillations.
         • Amplitude
           • Difference between avg. and max. pop. size. Directly proportional to fluctuation size.
         • Period
           • Time taken for one complete cycle to occur. Always about 4t (time lag). Mammals in high-latitudes cycle with peaks every 4 years - t = 1 year (reproduce once a year) (May 1976).
     • dN/dt = rN (1-N/K)
       • (1-N/K) = proportion of carrying capacity which is unused. If <1 = Growth If >1 = Decline Will always resolve to carrying capacity.
     • Maximum Sustainable Yield
       • N = 0.5k = high growth rate. Recruitment rate = proportion of first or second year animals in the population.
       • Peruvian Anchovy Fishery (Deriso 1987)
         • Fish quota maintained population at 0.5K. However, El Nino event (1972) led to decreased recruitment rate so when population was reduced re: quota the population never recovered.
     • Pheasants of Protection Island (Newcomb 1940)
       • 1937: 8 pheasants introduced. Abundant food, no predation, no migration. 1939: 40 pheasants 1942: 2000 pheasants Predicted curve 3x greater than actual - due to depleting resources (k) reducing growth rate.
   • Nt+1 - Nt
     • Non-overlapping change equation. Population one generation after a given time - current population = population change.
     • λ
       • Population change as a proportion of current population. λ = 1.2 = 20% growth.
   • r
     • Intrinsic (average) rate of growth of a species. Determined by life history characteristics. Time lags are inversely proportional to r. Faster growing organisms (invertebrates) show population dynamics in shorter spatial scales (May 1976). Populations with higher r values will track changes in k better.
     • Var. r > 2x r
       • Extinction
       • Grizzly Bears of Yellowstone (Gotelli 2001)
         • Est. r = -0.003034. Implies eventual declines over long enough. However, large variance of r. Increase could occur, or extinction within 200 years.
     • Doubling Time
       • Time required for population to double in size. Directly proportional to r. Exponential growth = doubling time at regular intervals.

2. Competition

   • Competitive interactions = two species negatively influencing each others' growth rate (suppress pop. size). Exploitation = Use of a shared resource Interference = Suppress exploitation ability Apparent = Effect of species A on B, transmitted through species C.
   • Lotka-Volterra Interaction Model
     • Each species shows logistics growth. Each growth rate hindered by presence of other pop.
     • α / β
       • Competition coefficients. a/b = 0 = no effect (single sp.). a/b = 1 = interchangeable. a/b = 4 = one individual of a = 4 individuals of b.
     • N1 = (K1 - aK2) / (1-ab)
   • Competitive Exclusion
     • Extirpation due to competitive interactions from another species. 
   • Intraguild Predation
     • Competitors also predator and prey. y/delta = interaction coefficient.  Doesn't alter carrying capacities
   • Intertidal Sandflat Worms (Gallagher et al. 1990)
     • Isoclines = H. florida should exclude others. However, coexistence. Diatom abundance varies with season - isoclines so close there isn't enough time for one species to become excluded due to growth rates.
   • Gerbil Competition (Abramsky et al. 2001)
     • One sp. better competitor (larger). Habitat divided to exclude larger individual in half. Decreased activity of smaller sp. when larger sp. present. Intermediate = no decreased activity due to shift to habitat without larger sp. Low/high densities = competition has larger sp. either needs to compete more or is more abundant.

3. Movement

   • Metapopulations
     • Populations consisting of smaller, local populations which are connected by migration. Distances are further than individuals will travel, but not can travel.
     • Sources
       • High-quality habitat which produces emigrants
     • Sinks
       • Low-quality habitat, which cannot sustain a population without immigration. Gutiérrez (1996).
     • Pt
       • Fraction of patches occupied at a given time by a population. dp/dt = colonisation - extinction Persistence = colonisation > extinction
       • m
         • Colonisation rate. Directly proportional to the number of empty patches. Increased patches = increased population.
     • Northern Spotted Brown Owl (Noone & McKelvery 1996)
       • Pop. declines = 7%/year 1950: 10-15% of original forest was left. <20% leads to pop. collapse in territorial animals. HCAs = Habitat conservation areas. Metapop. model used to show why habitat had to be protected. Indicator species.
     • Silver-Studded Blue Butterfly (Begon et al. 1996)
       • 9 metapopulations in N. Wales. Individuals don't cover very large distances (Suitable habitat <1km away). Dispersal distances intermediate in relation to the dist. of suitable habitat. 
     • Habitat Destruction
       • Levin's Rule
         • N suitable patches after destruction > N suitable patches unoccupied before disturbance Hanski et al. 1996b
       • Minimum Viable Metapopulation
         • Lowest number of local populations required for a metapopulation to persist into the foreseeable future. Minimum amount of suitable habitat required (MASH).
       • Rescue Effect (Brown & Kodric-Brown 1997)
         • Increasing immigration causes increasing population size, which reduces the likelihood of extinction. More occupied patches = greater immigration.
     • Multi-Species
       • Competition
         • Inferior competitor can persist if m:e > m:e for superior competitor. Can relocate to empty suitable patches to avoid local extinction (competitive exclusion).
   • Brownian Motion
     • Finite area = mean displacement increases linearly with time Displacement increases linearly with number of moves.
     • Composite Brownian Motion (Jansen et al. 2012)
       • Random movements in a series of spatial scales - not power log. Important for invasive sp. to expand from a frontier.
   • Lévy-Type Movements
     • Cover space much more quickly - increased pop. spread. When target sites are sparse or total area is 'infinite' (very large).
   • Species-Distance Relationship
     • Areas further from source support fewer species. Species-Area relationship = larger areas support more species. Darlington's Rule: A 10x increase in the size of area will result in a 2x increase in sp. number.
   • Moran Effect (Jansen 1999)
     • Two populations, regulated by the same density dependence and env. conditions become synchronous in pop. cycles due to env. disturbance.  
   • Phase-Locking
     • Two populations couple via dispersal = synchronicity. Weakens as distances between populations increases.
   • Spatial Systems & Predation (Huffaker 1958)
     • Oscillations asynchronous across heterogeneous habitat. Patches reduce predator success. Predators won't colonise empty patches (no prey), but prey will (resources). Allows prey to grow populations before detection and reduction.
     • Habitat Structure & Mite Predation (Ellner 2001)
       • Local extinctions = resource exhaustion + predation. Variation comes from habitat structure, not predation. Reduction in predator success from 90 - 68%. Plants consumed asynchronously and so are prey outbreaks.

4. Predation

   • Trophic interactions = one sp. using another sp. to feed and reproduce on
   • Functional Response
     • Measures per predator effect of predation of prey's growth rate. Greater = greater depression on prey's growth rate.
     • Type 1
       • Each predator consumes more prey as prey densities increase. Functional response vs prey density, x = capture efficiency. Unrealistic: Limited by handling time.
     • Type 3
       • Feeding rate accelerated at low prey densities. Occurs when predators; switch to more common prey or develop a search image
     • Type 2 (Holling 1959)
       • Total foraging time = time searching for prey + handling time. Maximum prey consumption limited by the physical capacities of a predator to handle its prey Larger prey = predation rate plateau at lower densities
   • α
     • Capture efficiency
   • q
     • Predator death rate
   • β
     • Conversion efficiency. Ability of predator to convert each new victim into additional per capita growth rate.
   • Alternative Isoclines
     • Predator
       • Carrying capacity curves isocline to the right. Predators can also alter their food source if prey populations become scarce (Predator switching).
     • Prey
       • Increased at low densities. Constant prey immigration seasonally OR refuges. Total refuge = protects a given number. Effective refuge = protects a given proportion.
   • Evolutionary Context
     • Evolution preferentially selects specialists as they have a greater capture efficiency - can increase their fitness the most for the least. Carnivores receive greater nutrition than herbivores (amino acids) - can eat at a lower rate for same body mass.
   • Lynx & Snowshoe Hare (Keith 1983)
     • Hare populations cycle with peak abundance every 10 years. Lynx populations highly synchronised - peak 1-2 years after hares. Increased hare pop. = increased grazing = increased toxins in grass = decreased hare pop.

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