Organisms and Populations Set-5

Q1. A population grows exponentially with intrinsic rate of increase $r=0.02\ \text{day}^{-1}$ and initial size $N_0=500$. Using $N(t)=N_0e^{rt}$, after approximately how many days will the population reach $1000$ individuals?

Q2. A life table for a species gives number alive at start of age 0 = 100, at start of age 1 = 40, and at start of age 2 = 20. Fecundity (daughters per female) is $m_1=1.5$ at age 1 and $m_2=2.5$ at age 2. Using $R_0=\sum l_x m_x$ with $l_x$ as proportion surviving to start of age $x$ (take $l_0=1$), what is the net reproductive rate $R_0$?

Q3. Consider a population with net reproductive rate $R_0=0.9$. Evaluate the statements:
(I) The total population size will necessarily decrease in the next few years.
(II) If the current age structure is heavily weighted toward reproductive-age individuals, the total population may still increase in the short term despite $R_0<1$.
Which option is correct?

Q4. A species exhibits a strong Allee effect; its deterministic dynamics can be approximated by

with $K=1000$ and critical threshold $N_c=50$. Two identical reserves are stocked with founders: Reserve A with $40$ individuals and Reserve B with $60$ individuals. Assuming no immigration and deterministic dynamics, which outcome is most likely?

Q5. In a local population per capita birth rate is $b=0.40\ \text{yr}^{-1}$, death rate $d=0.25\ \text{yr}^{-1}$, immigration per capita $i=0.02\ \text{yr}^{-1}$ and emigration per capita $e=0.05\ \text{yr}^{-1}$. For initial population $N_0=1000$ and ignoring density dependence, what is the expected population size after one year? (Use $N_1=N_0(1+r)$ where $r=b-d+i-e$.)

Q6. A 2000 m$^2$ grassland is sampled using five 1 m$^2$ quadrats. Counts of a plant species in the quadrats are 7, 9, 8, 6 and 10. Using the quadrat estimator $\hat{N}=\bar{n}\times\frac{A}{a}$ (where $\bar{n}$ is mean count per quadrat, $A$ total area and $a$ quadrat area), estimate the total number of individuals in the grassland.

Q7. A population follows logistic growth given by $\displaystyle \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)$. For $r=0.6\ \text{yr}^{-1}$ and $K=2000$, calculate the instantaneous growth rate $\frac{dN}{dt}$ when $N=1600$ and state the immediate trend.

Q8. A wetland is surveyed using the Lincoln–Petersen mark–recapture method. In the first visit $M=120$ frogs were captured, marked and released. After mixing, a second visit captured $C=150$ frogs, among which $R=30$ were marked. Using $\displaystyle \hat{N}=\frac{MC}{R}$, estimate the population size and state how loss of marks (some marked frogs losing their marks before recapture) would bias the estimate.

Q9. For a population with age classes $x=0,1,2,3$ the survivorship $l_x$ and fecundity $m_x$ are: $l_0=1.00,\ m_0=0;\ l_1=0.60,\ m_1=1.0;\ l_2=0.30,\ m_2=1.5;\ l_3=0.05,\ m_3=0.5.$ Compute $R_0=\sum l_x m_x$, $T=\dfrac{\sum x l_x m_x}{R_0}$ and approximate intrinsic rate $r\approx\dfrac{\ln R_0}{T}$. Which choice best represents $r$ and the population trend?

Q10. A model with a strong Allee effect has per-capita growth rate $g(N)=r\left(1-\dfrac{N}{K}\right)\left(\dfrac{N}{A}-1\right)$, so $\dfrac{dN}{dt}=g(N)N$. Let $r=1\ \text{yr}^{-1}$, $K=1000$, $A=50$. If the initial population is $N_0=30$, what is the most likely fate of the population?

Q11. In a grassland study, 30 individuals of a beetle species were captured, uniquely marked and released. A week later 45 individuals were captured, of which 15 were marked. Using the Lincoln–Petersen estimator $N = \dfrac{n_1 n_2}{m}$, estimate the population size.

Q12. A population follows discrete geometric growth $N_t = N_0 \lambda^t$. If $N_0 = 50$ and $N_4 = 162.5$, compute $\lambda = \left(\dfrac{N_4}{N_0}\right)^{1/4}$ and predict $N_6$ (round to nearest integer).

Q13. For a species the life table values are: age $x = 0,1,2,3$; survivorship $l_x = 1.00,\;0.60,\;0.30,\;0.05$; fecundity $m_x = 0,\;2,\;4,\;1$. Using $R_0=\sum l_x m_x$, $T=\dfrac{\sum x l_x m_x}{R_0}$ and $r \approx \dfrac{\ln R_0}{T}$, which statement is correct?

Q14. In the Levins metapopulation model the equilibrium fraction of occupied patches is $p^* = 1 - \dfrac{e}{c}$ where $c$ is colonization rate and $e$ extinction rate. If initially $c=0.30\ \text{yr}^{-1}$ and $e=0.15\ \text{yr}^{-1}$, and fragmentation reduces $c$ to $0.15\ \text{yr}^{-1}$ while $e$ remains unchanged, which of the following is correct?

Q15. A fish stock follows $\dfrac{dN}{dt} = rN\left(1-\dfrac{N}{K}\right) - H$, where $H$ is constant harvest. Maximum sustainable yield is $MSY=\dfrac{rK}{4}$ obtained at $N=\dfrac{K}{2}$. For $r=0.8\ \text{yr}^{-1}$ and $K=10\,000$, calculate MSY and state the likely outcome if managers set a constant annual harvest $H=2\,500$.