Organisms and Populations Set-2

Q1. A population of rodents has intrinsic rate of increase $r=0.2\ \text{yr}^{-1}$, current population size $N=100$, and carrying capacity $K=500$. Using the logistic growth equation $\displaystyle \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$, the instantaneous population growth rate $\frac{dN}{dt}$ (individuals per year) is:

Q2. In a mark–recapture study the Lincoln–Petersen estimator is $\displaystyle \hat{N}=\frac{n_1 n_2}{m_2}$. A researcher marks $n_1=100$ beetles and later captures $n_2=80$, of which $m_2=10$ are recorded as marked. It is later discovered that 20% of originally marked beetles had lost their marks before the second sampling (so some recaptured marked individuals were not recognized). Compared to the true population size, the estimated $\hat{N}$ computed from the observed $m_2$ will most likely be:

Q3. The life-table for a species gives age-specific survivorship $l_x$ and fecundity $m_x$ as follows: age $x=1:\ l_1=1.0,\ m_1=0$; $x=2:\ l_2=0.6,\ m_2=2$; $x=3:\ l_3=0.2,\ m_3=1.5$. Using $R_0=\sum l_x m_x$, $T=\dfrac{\sum x l_x m_x}{R_0}$ and $r\approx \dfrac{\ln R_0}{T}$, the intrinsic rate of increase $r$ (per year) is approximately:

Q4. In the Levins metapopulation model the equilibrium fraction of occupied patches is $p^*=1-\dfrac{e}{c}$, where $c$ is the colonization rate and $e$ the extinction rate. If $e=0.2\ \text{yr}^{-1}$, what minimum colonization rate $c$ (in yr$^{-1}$) is required to ensure at least $70\%$ ($p^*\ge 0.7$) of patches remain occupied at equilibrium?

Q5. Two species interact via Lotka–Volterra competition. Competition coefficients are $\alpha_{12}=0.4$ (effect of species 2 on 1) and $\alpha_{21}=0.6$ (effect of species 1 on 2). Their carrying capacities in isolation are $K_1=200$ and $K_2=150$. The model predicts stable coexistence if $K_1>\alpha_{12}K_2$ and $K_2>\alpha_{21}K_1$. Given these values, the expected long-term outcome is:

Q6. A population grows continuously according to the model $N_t = N_0 e^{rt}$. If $r = 0.693\ \text{day}^{-1}$ and the initial population $N_0 = 50$, the population size after $3$ days is closest to:

Q7. Using the life-table values below, calculate the net reproductive rate $R_0 = \sum l_x m_x$ and choose the correct interpretation.

Age $x$: 0, 1, 2, 3
$l_x$: 1.0, 0.6, 0.3, 0.05
$m_x$: 0, 1.2, 2.0, 1.6

Q8. A field study sampled 10 equal quadrats and recorded individuals per quadrat: $\{2,0,3,1,4,0,2,3,1,4\}$. Using the index of dispersion $ID = \dfrac{s^2}{\bar{x}}$ (where $s^2$ is variance and $\bar{x}$ the mean), which dispersion pattern does this data indicate?

Q9. Assertion (A): An Allee effect can create a critical population threshold below which a population tends to decline to extinction even when resources are abundant.
Reason (R): This effect occurs solely because predators preferentially target low-density populations, increasing per-capita mortality at small population sizes.

Q10. In the Lotka–Volterra competition model the zero-growth isoclines are $N_1 + \alpha_{12}N_2 = K_1$ and $N_2 + \alpha_{21}N_1 = K_2$. For two species the parameters are $K_1 = 400,\ K_2 = 300,\ \alpha_{12} = 1.2,\ \alpha_{21} = 0.8$. Based on these values, which equilibrium outcome is predicted?