Chapter 54, Problem 1Q

Summary Introduction

To analyze:

The net reproductive rate (R₀) for population and investigate its value.

Given:

An incomplete cohort life table is provided in Table 1, from which the components given in columns 4 and 5 are to be calculated.

Table 1: A life table of Australian sharp nose shark showing survivorship and fecundity.

Age (x)	Survivorship of females (lx)	Fecundity of female offspring (mx)	lx mx	x lx mx
0	1	0
1	0.32	1.98
2	0.19	2.59
3	0.11	2.81
4	0.06	2.89
5	0.03	2.92
6	0.02	2.93
7	0.01	2.93
8	0.005	2.93
9	0.003	2.93
10	0.002	0

Introduction:

A cohort life table shows demographic changes in a population. The table provides information about rate of reproduction, generation time, fecundity, rate of survival of the individuals and all information required to study a population in detail.

Explanation of Solution

The survivorship (l_x) of Australian sharpnose shark is calculated by dividing number of surviving individuals by number of individuals that were present at the time of birth (N₀). The fecundity (m_x) of an individual female is calculated by dividing the number of female offspring (N_x) by the number of females who have produced these offspring (N_x offspring).

A number of offspring per age class are calculated by simply multiplying l_x with m_x. The value of l_xm_x for the age class 1 is calculated as follows:

$\begin{array}{c}{\text{l}}_{\text{x}}{\text{m}}_{\text{x}}=\text{1}\times \text{0}\\ {\text{l}}_{\text{x}}{\text{m}}_{\text{x}}=\text{0}\end{array}$

The values of l_xm_x for the subsequent age classes can be calculated by using a similar method.

Table 2: A complete life table of Australian sharpnose shark.

Age (x)	Survivorship of females (lx)	Fecundity of female offspring (mx)	lx mx
0	1	0	0
1	0.32	1.98	0.63
2	0.19	2.59	0.49
3	0.11	2.81	0.31
4	0.06	2.89	0.17
5	0.03	2.92	0.08
6	0.02	2.93	0.06
7	0.01	2.93	0.03
8	0.005	2.93	0.014
9	0.003	2.93	0.008
10	0.002	0	0

From Table 2 net reproductive rate (R₀) of the shark population can be calculated by using,

$\begin{array}{c}{\text{R}}_{\text{0}}=\text{sum }\left({\text{l}}_{\text{x}}{\text{m}}_{\text{x}}\right)\\ {\text{R}}_{\text{0}}=\text{1}\text{.792}\end{array}$

The net reproductive rate for the shark population is 1.792 that is greater than 1. So, it represents a net increase in the number of offspring in each generation. The population growth here is exponential.

Conclusion

Therefore, it can be concluded that reproductive rate of the shark population is 1.792, which indicates the exponential population growth and increase in offspring in each generation.