Ostrich Farm Risk Analysis Using Lefkovitch Matrix Report

This report presents a risk analysis for a hypothetical ostrich farm, utilizing the Lefkovitch matrix and MATLAB software to model population dynamics. The analysis aims to determine the survival rate of the ostrich population, the impact of human activities, and the maximum sustainable harvesting rates for eggs and meat. The report addresses the challenges of managing an ostrich farm, where only a small percentage of the population are breeders. It constructs a Lefkovitch matrix based on provided data to compute the finite rate of increase, project population sizes, and optimize harvesting strategies for maximum income. The discussion highlights the importance of breeder management for farm continuity and the need for balanced harvesting to sustain both meat and egg production. The report concludes with a summary of the findings, emphasizing the critical role of breeders and the necessity of a comprehensive management strategy to mitigate risks and ensure the farm's long-term viability.

ABSTRACT
This paper prepares a report for a hypothetical risk analyst whose job is to decide whether to
approve an application made for a bank loan. The lab tests are conducted using the MATLAB
software. Ostriches are considered wild birds but they can be reared domestically for their
products. These birds yield eggs, good meat, leather, and plenty of feathers. From experience, it
has been noted that the skin quality of the ostriches decreases with the increase in age. The
ostriches tend to be slaughtered before they start breeding as some of the birds are left in the
fields as key breeders (Caughley, n.d.). The larger number is slaughtered. This paper seeks to test
and develop a model of an ostrich farm.
INTRODUCTION
The Lefkovitch matrix is an effective tool used to determine the growth of a population as well
as the stage number distribution within the population over time. This matrix is the correct
approach for animals but it is not highly utilized in the analysis of the plant population. The
elements of the matrix do not represent proportional survivorships rather they are the
probabilities of making transitions from one stage to another. The Lefkovitch matrices describe
populations with stages or size structures (Starfield & Bleloch, n.d.).
OBJECTIVES
(a) To determine the survival rate of the ostrich population from a given Lefkovitch matrix.
(b) To determine the impact of human activities on the survival rate of the population.
(c) To determine the maximum rate of harvesting eggs and meat from the birds based on the
Lefkovitch Matrix model.
PROBLEM STATEMENT
There is a great need to have a good management strategy in the ostrich farm. Previous managers
have not managed to do a good job. The number of ostrich populations per stage stabilized at
shares as if ostriches lived in the wild. It was observed that among the entire population of birds,
only 15 percent were breeders. This was not a good trend for an ostrich farm that requires
continuity by ensuring larger breeding chances. The rest ended up being slaughtered for meat
and skin.
ASSUMPTIONS
(i) The stage-structure model is a probabilistic model. This is demonstrated such that the
number of eggs or birds in a stage is not required to be a whole number. For instance,
if according to the model there are 57.7 baby chicks after 7 months, it means that the
actual number may be bigger or smaller but 58 is more likely than 57.
(ii) When we say that a certain harvesting plan stabilizes the ostrich population, it does
not mean that every month populations will be exactly like in the previous month.
They may grow a little but whenever they do, we can just harvest a little bit more to
get them back to normal.

LAB RESULTS
The current statistics on the ostrich farm is as shown in the table illustrated below,
Stage 0 1 2 3 4 5 6 7
Population 40 28 25 22 40 50 30 40
Share in
%
14.5455 10.1818 9.0909 8.0000 14.5455 18.1818 10.9091 14.5455
Question 1
Constructing a Lefkovitch matrix L from the data above.
Theoretically, the matrix L is formulated by obtaining the computation of P, F, and λ for a
population that exhibits stage structure. We aim at computing λ which is the finite rate of
increase for the population. In the matrix model, it is possible to compute the time specific
growth rate by rearranging the term such that,
Assuming that there are five key stages in the Ostrich life that the Ostrich farm owners or
management are concerned with it is possible to obtain the sum total of the initial ostrich
population. The population includes the eggs considering about 36 months.
L= [1.6 1.5 0.25 0.1; 0.8 0 0 0; 0 0.5 0 0; 0 0 0.3 0];
At year 36, the populace’s finite rate of upsurge is 0.8. the steadied growth is readily apparent by
examining the semilog graph, where the projection lines for each stage class become parallel to
each other. The trend at the 36th year and some few preceding and succeeding years demonstrate
a level of constant growth trail. At the stable dispersal, the population is subjugated by the baby
chicks, the growers, the finishers, and the slaughter birds. For all these stages the λ = 0.8.
The output of the matrix is

Plot of the monthly breeders for 3 years ahead, that is plot Xt7 for t=0, 1, 2, …, 36.
Plotting on a semi log graph,
semilogy(Gen,NPop); xlabel('Generation'); ylabel('Population Size');title('Lefkovitch Model');
legend('Starters','Growers','Finishers','Slaughter Birds','Breeders')
[V,D]=eig(L);
norm=sum(V(:,1));
age=V(:,1)./norm;
The output is obtained as shown below,
Question 2

The projected population size is obtained by
Nt=N 0× λt
This assumes that the finite rate of increase in the population growth rate is invariant over time.
>> k=mean(g);
>> k'
ans =
0.3424
0.4861
0.3375
0.3375
0.5436
0.1248
0.0263
0.0263
Question 3
Harvesting meat and skin
The solution seeks to determine the largest real percentage as a whole number of the slaughter
birds stage. These birds are projected to be slaughtered each month if the model is implemented
on the Ostrich Farm.
n t c xi i
t
i
i
N
( )  
 
1
Where the C values are the constants set by the initial population structure.
population of slaughter birds=50
The number of slaughter birds eventually slaughtered , 18.1818 %of 50 birds
¿ 18.1818
100 ∗50=9.0909initially
With a growth rate of λ = 0.8, we obtain,
¿ 9.0909
0.8 =11.36∨12 slaughter birds per season

Question 4
Optimizing the harvest for maximum income
The plan is to harvest both eggs and birds. The information provided seeks to have the model
implemented over the ostrich farm. The total number of the birds in the farm at this point is 500.
Considering that one bird yields a net income of $260, one can figure out the percentage of the
eggs harvested as well as the birds slaughtered. It is also known that selling one egg costs the
buyer around $21.
Eig(L) is obtained from the above equation
λ = 0.3256.
DISCUSSION
The mathematical model used to develop the model of an ostrich farm is the Lefkovitch matrix.
The ostriches are classified into eight stages namely the eggs, the baby chick or the pre-starter
stage, the starter, the growers, the finishers, the slaughter bird level, maintenance bird, and the
breeder. The slaughter bird level has all the birds that are slaughtered for meat and skin. The
breeders are mandated to procreate by laying eggs to start the life cycle. In the mathematical
model, the different stages are labelled 0 to 7 respectively. The time is discrete and a unit interval
of one month is used in the model design. The matrix elements correspond to the transition rates
or probabilities between stages as opposed to simply the survival and fecundity. These transition
rates depend in part on survival rate but also on growth rates (Clark, n.d.).
CONCLUSION
In a nutshell, the risk portfolio above shows the information required to determine the risks
involved when a certain stage of the birds is not sustained. The breeders need to be looked after
as they determine the continuity of the farm. The slaughter birds need to be well nurtured to
produce more meat and skin hence more sales.
REFERENCES
Caughley, G., n.d. Mathematical Ecology. 2nd ed. New York, USA: John Wiley & Sons.
Clark, C. W., n.d. Mathematical models in the economics of renewable resources. SIAM Review,
1(21), pp. 100-117.

Starfield, A. M. & Bleloch, A. L., n.d. Building Models for Conservation and Wildlife
Management. New York, USA: Macmillan Publishing Company.