Chapter 2 Lecture. Density dependent growth and intraspecific competition ~ The Good, The Bad and The Ugly. Spring 2013

Transcription

1 Chapter 2 Lecture Density dependent growth and intraspecific competition ~ The Good, The Bad and The Ugly Spring 2013

2 2.1 Density dependence, logistic equation and carrying capacity dn = rn K-N Dt K Where K = carrying capacity = # of individuals that can be maintained indefinitely. Because most populations do not sustain exponential growth for long periods of time. Fig. 2.1 Fig. 2.2 Per capita growth rate = divide growth between time intervals t and t + 1 by the pop size at t+1.

3 Density dependence, logistic equation and carrying capacity dn = rn K-N Dt K Logistic model is based on competition ~ competitive interactions for a resource in short supply Competition by definition results in a reciprocal negative interaction between the individuals competing -/- Intraspecific Competition Interspecific Competition

4 Density dependence, logistic equation and carrying capacity dn = rn K-N Dt K Interference competition (Park 1962) = access to a resource is prevented, ~blocked or prevented Ex., Depletion competition ~ exploitative competition (Park 1962) = one group is better at utilizing the resource without direct interference Ex., Intraspecific competition occurs through density dependent responses in birth, death, and growth rates (r and λ) and adult size (esp. with determinant life cycles).

5 2.2 Density dependence with discrete generations Begin with eq. 1.4: N t = N 0 R t To include intraspecific competition need to modify net reproductive rate = R Fig. 2.3 Assumes EXACT density dependence One approach: Graph reciprocal of increase per generation (N t /N t+1 ) versus N t Then: population can grow at its maximal rate. R. (N t /N t+1 )/N t = 1/R Therefore: Point A = (0,1/R) Point B = (K, 1) Assumes a straight line would connect the two points ~ exact density dependence = exactly compensating for density dependence REALISTIC? Carrying capacity is reached when (N t /N t+1 )/N t = 1

6 2.2 Density dependence with discrete generations Eq 2.2a simplifies eq. 2.1 With a = (R-1)/K ~ carrying capacity parameter Then N t+1 = N t R/1 + a N t R I = Density independent growth R A = Density dependent or actual growth *** Thus if N is small (or far below K), then R A is more similar to R I until K is reached. See Table 2.1, pg 39

7 Therefore: To relax assumption of linear density dependence: replace -1 with b* -1 R A = R I 1 + N t R I N t K Linear (eq. 2.3b) R A = R I 1 + N t R I N t K -b* Non-Linear (eq. 2.4)

8 b* non linear approach Exact compensation: b* = 1 with slope -1 Overcompensation: b* > 1 with slope < -1 Undercompensation: b* < 1 with pop size decline slower than expected. Figs. 2.4 and 2.5 pg. 40 -b* R A = R I 1 + N t R I N t K

9 Law of constant final yield Defn: Total yield is constant but density & plant size vary Fig. 2.6 pg. 41 Commonly used in botanical & agric. research C = Nw Where: C = final constant yield, kg/area N = density of plants/animals per unit area w = mean mass per organism in kg

10 C = Nw m (1+a N) b* (eq 2.7) C = Law of constant yield ~ Self thinning rule N = density w m = maximum potential mass per plant w= actual mean mass of organism a = carrying capacity parameter Self-thinning properties: i) indiv s grow and increase in size ii) iii) When critical density reached, e.g., self-thinning limit, mortality begins Pop reaches stage where increase in mass of some indiv. results in mortality of others

11 2.3 Density dependence - overlapping generations Starting with the logistic differential equation: Eq 2.8 dn/dt = rn K-N K Eq. 2.9 r a = r m K-N K r a = actual growth as modified by K r m = maximum growth without competition

12 Assumptions of Logistic Equation 1) Carrying capacity (K) is constant 2) Pop growth unaffected by age distribution 3) b and d rates change linearly with pop size (Fig. 2.9) 4) interaction between pop and K is instantaneous 5) abiotic density indep. factors that do NOT influence b and d rates 6) crowding affects all members of community in a similar fashion **Assumptions can be met in lab but not in the field. Consequences? Fig. 2.8

13 2.4 Assumption 3 = Non-linear density dependence of b and d rates and Allee effect Allee Effect (1931) proposed that many species have a minimum population size (MVP) At which basic services are not reliably provided ~ pollinator service for small plant populations (Groom, 1998). Small populations have a higher probability of extinction. WHY? 1) Group cooperation reduces loss from predators 2) Group foraging is beneficial 3) Small poplns more susceptible to density independent or stochastic events

14 N Ne ~ effective number Inbreeding depression Population size influences Small Large < s # of reproductive individuals greater less Genetic drift greater Less Gene flow Less likely More likely Genetic variability less greater Fig non linear response to b and d rates to pop density Fig Allee effect and MVP Fig Effect of non linear feedback on logistic growth

15 Nonlinear modifications to logistic equation ~ feedback w/r Assumption 3 Theta (θ ) logistic model helps establish the Ricker equation and is important because it is the basis for other population models such as those used to describe predator-prey interactions. Distinguish between r A and r m (actual vs. maximum growth) N t+1 = N t e r(k-nt/k) = N t e r(1-nt/k) (eq Ricker equation 1952) Replace θ as a superscript for (N/K) such that when: Θ = 1.0, results in traditional logistic growth to density Θ < 1.0, results in dens dep strong even though pop far below K Θ > 1.0, results in dens dep weak until pop close to K N t+1 = N t e r(1-(nt/k)θ) Eq Fig and 2.14 examples

16 2.5 Time lags & limit cycles feedback w/r Assumption 4 For Continuous populations (eq. 2.16) dn/dt = rn t (K-N t-τ / K) For Discrete populations (eq. 2.17): N t+1 = N t e r(k-nt-t/k) The popln responds to K based on pop size at tau (T) = time units in the past Time lag along with r produces interesting patterns Product of r and T determines behavior of population ~ lag effects Fig 2.15

17 2.5 Time lags & limit cycles feedback w/r Assumption 4 Product of r and T determines behavior of population Fig 2.15 pg. 53 rt between 0 and 0.37 follows logistic eq. and pop achieves stable point rt between 0.37 and 1.57 temporary oscillations but does stabilize rt between 2 and 1.57 permanent oscillations around K = limit cycle rt > 2.0 oscillations extreme, popln goes extinct

18 2.6 Chaos & behavior of discrete logistic model Time lags are naturally understood with discrete pops, remove Tau and we return to Ricker equation (Fig. 2.16, stable equil.; r < 2.0) N t+1 = N t e r(k-nt/k) = N t e r(1-nt/k) However, if growth rate is large (r or λ), product of rt determines the behavior of the population. Fig (2.0 > r < 2.53 = discrete 2 point cycle) Fig (2.53 > r < 2.66 =discrete 4 point cycle) Fig When r > 2.69 then Chaos experienced in discrete logistic model Table 2.3 Relationship between r, R, λ & behavior of discrete logistic model. Stable, 2-point, 4-point, 8 point and chaos.

19 2.7 Adding stochasticity to density dependent models Environmental stochasticity Demographic stochasticity The more variability experienced by a population, the smaller the population is expected when compared to a deterministic model. Combined effects of environmental and demographic stochasticity increases the {prob. of extinction} Figs Deterministic vs. stochastic growth: Pop size Fig Effect of demographic stochasticity (r) and environmental stochasticity (K) on pop growth Time

20 Population density increase leads to: 1) A linear increase in mortality 2) A linear decrease in fertility 3) A reduction in average growth rate 4) A reduction in average adult size

21 Laboratory and Field data What insights can you provide from the examples discussed in Chap. 2 and the models from Chap. 1 & 2? Frogs, Rana tigerina Harp seals, Phoca groenlandica Limpets, Patella cochlear Ant colonies, Pheidole pallidula Horned beetles, Dung, genus Onthophagus Little brown bird, Dunnock, Prunella modularis Lions, Panthera leo

22 Chap 2 Highlights Density dependence in poplns with discrete generations Density dependence in poplns with overlapping generations Nonlinear density dependence of birth and death rates Allee effect Time lags and cycle limits Chaos and behavior of discrete models Adding stochasticity to density-dependent models Laboratory and field case studies Behavioral aspects of intraspecific competition