Exercises: Carrying Capacity, Limiting Factors and Growth Curves

Define the term carrying capacity of an environment. Distinguish between a density-dependent and a density-independent limiting factor, giving one example of each for a population of red deer in Killarney National Park.

A population of bacteria grows in a flask of nutrient broth. Name the shape of the growth curve (S-shaped or J-shaped) shown by a population that grows exponentially with no limits, and name the shape shown by a population that levels off. State which of the two curves is more commonly seen in natural populations over a long period.

The number of rabbits on an island off the Cork coast is recorded each year.

1. a) Name the type of growth shown.

1. b) Calculate the percentage increase in population from year 1 to year 2.

1. c) State two factors that would eventually limit this growth.

A population of 50 grey squirrels is introduced to a woodland in County Wicklow. The population size is recorded every two years:

1. a) Describe the shape of the curve and identify the phases of growth (lag, exponential, stationary).

1. b) Estimate the carrying capacity of the woodland.

1. c) Calculate the average rate of increase in squirrels per year between year 4 and year 6.

1. d) Suggest two limiting factors responsible for the levelling off.

Zebra mussels, an invasive species, colonise a stretch of the River Shannon. In the first year there are 1,000 mussels per square metre, and the population increases by 60% each year when resources are plentiful.

1. a) Calculate the density after 3 years of unrestricted growth.

1. b) Explain why this J-shaped growth cannot continue indefinitely.

1. c) Explain what is meant by a density-dependent factor and give two examples relevant to this population.

A farmer in County Meath stocks a field with sheep. When 10 sheep are kept, each gains 8 kg per season. When 30 sheep are kept, each gains 5 kg per season. When 60 sheep are kept, each gains 1 kg per season, and when 80 sheep are kept some lose weight.

1. a) Calculate the total weight gain of the flock at 10, 30 and 60 sheep.

1. b) Explain these results in terms of carrying capacity and competition.

1. c) Suggest, with a reason, a sensible stocking level for the farmer and how the carrying capacity might be raised.

A population of lynx and a population of snowshoe hare (a predator and its prey) are monitored in a boreal forest. The hare population peaks, and the lynx population peaks about two years later, followed by a crash in hares and then in lynx.

1. a) Explain how predation acts as a density-dependent limiting factor for the hares.

1. b) Explain the time lag between the two peaks.

1. c) Discuss whether the carrying capacity of the hares is a fixed number, with reference to food supply and predation.

1. d) Describe how this pattern differs from a simple S-shaped curve.

The logistic growth of a population is described by $\frac{dN}{dt}=rN\left(\frac{K-N}{K}\right)$, where $N$ is population size, $r$ is the intrinsic rate of increase and $K$ is the carrying capacity. A population of fish in a lake has $r = 0.4$ per year and $K = 5,000$.

1. a) Calculate $\frac{dN}{dt}$ when $N = 500$, $N = 2,500$ and $N = 4,900$.

1. b) State at which population size growth is fastest and explain why.

1. c) Explain the biological meaning of $\frac{K-N}{K}$ and what happens to growth when $N = K$.

1. d) A fishing quota is to be set to give a sustainable yield. Using your results, recommend a population size to maintain and justify your answer.

Red squirrels in Ireland are declining where grey squirrels have been introduced. Both species compete for food and the grey squirrel also carries the squirrelpox virus, to which it is immune.

1. a) Explain, using the concept of carrying capacity, why the red squirrel population falls in areas shared with greys.

1. b) Identify two limiting factors affecting the red squirrel and classify each as density-dependent or density-independent.

1. c) Explain why a disease may act differently on the two species.

1. d) Propose two management measures to protect red squirrels and justify them ecologically.