Population Dynamics

7730 /www/nrich/html/content/id/7730/ 1 2011-11-28T14:46:39 <mdoxml version="1.0"> <h3>Per Capita Rates</h3> It is important to relate the basic population parameters (such as births or deaths) to the size of the whole population. This allows us to make a better decision if a population is at risk. We define the per capita birth rate (or nativity rate) as the number of births per individual per unit time interval: $b=\frac {B}{N}$ . Similarly we define the per capita death rate (or mortality rate) as the number of deaths per individual per unit time interval: $d=\frac {D}{N}$ <h3> </h3> <h3>The First Model</h3> Recall the population equation from <a href="http://nrich.maths.org/7252?part=index">before</a>: $$N_{t+1}=N_t+B-D$$ Because per capita birth and death rates do not change with the size (or density) of the population, we can rewrite our model in terms of per capita rates: $$N_{t+1}=N_t+bN_t-dN_t=N_t+(b-d)N_t$$ This model is said to be density-independent. We call the term $r=b-d$, the geometric rate of increase. Note that $r=\frac{\Delta N_t}{N_t}$ , so r can be interpreted as the per capita rate of change of population size. The equation for our model becomes: $$\begin{align*} N_{t+1}&=N_t+rN_t \\ &=(1+r)N_t \\ &=\lambda N_t \end{align*}$$ where $\lambda=1+r$ is defined as the finite rate of increase. Note that $\lambda=\frac{N_{t+1}}{N_t}$ , so $\lambda$ can be interpreted as the ratio between the population size at one time to another time. <h3> </h3> How do you think we can solve this new equation? Go <a href="http://nrich.maths.org/7104?part=index">here</a> for more information. Do you think this model is valid in reality? What problems do you think might occur? Think about environmental resources and density-independence. An investigation of these problems can be found <a href="http://nrich.maths.org/7048?part=index">here</a>. Question: If 20 sea otters from a total population of 850 are fatally affected by disease, what is the mortality as a per capita rate? <mdo:image style="width: 300px; height: 200px;" src="seaotters1-300x200.jpg" alt=""></mdo:image> Given the population is initially 850, and increases to 1000 after one year, what is the value of $\lambda$? Use this to find the per capita birth rate, and find the population size in ten years. </mdoxml> 2 4 0 0 0 0 1 Population Dynamics - part 1 First in our series of problems on population dynamics for advanced students. Using, Applying and Reasoning about Mathematics Mathematical modelling Collections Population Dynamics Applications biology Pre-Calculus and Calculus Calculus generally