Report Prey Predator presentation

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The Effect of Sheep's Populations in the Presence of Coyote

Predator - prey Page 1

1.0 ABSTRACT

In this project, we are considering the evolution of a system of population

dynamics of prey and predator of two types of species. There are sheep as a prey and

coyote as a predator. This model will described by Lotka-Volterra Population Growth

Model. By using the Pplan8 and ODEsolve of Math lab, we can determine the graph and

the nature of stability of each model that we have modified. Jacobian matrix is used in

order to determine the nature of stability of critical point for each model. Based on the

result, the nature of stability for our basic model is stable centre and unstable saddle

point. On the other side, there are only three types of nature of stability for each model

that we have modified, which are unstable spiral, unstable saddle point and stable centre.


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2.0 INTRODUCTION

This project is about prey and predator model which the coyote as the predator and the

sheep as the prey. The Coyote is a canines species that smaller than wolves. It is also

known as prairie wolves, brush wolves or American jackal. Nowadays, it can be found at

the big cities such as Los Angeles and North America. The coyote once lived primarily in

open grassland and deserts, but now roam the continent's forests and mountains.

Coyotes are fearsome in the field where they enjoy keen vision and a strong sense

of smell. They can run up to 40 miles (64 kilometers) an hour. In the fall and winter, they

form packs for more effective hunting. In spring, females den and give birth to litters of

three to twelve pups. Both parents feed and protect their young and their territory. By the

following fall, the pups are able to hunt on their own. Coyotes usually breed only once a

year and 6 pups per birth in average. Coyote can live a maximum of 10 years in the wild

and 18 years in captivity.

The coyote is versatile feeders. Sometimes, it is labeled as carnivores but more

often as omnivores. They will eat almost anything. They hunt animals such as sheep,

rabbits, rodents, and frogs. They also happily dine on plants and fruits. Presently, the

coyote is the most abundant livestock predators in western North America, causing the

majority of sheep, goat and cattle died.When attacking adult sheep or goats, coyotes will

bite the throat just behind the jaw and below the ear, and they will bite the skull and

spinal regions when attacking smaller prey, such as young lambs.


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Meanwhile, sheeps are relatively small ruminants, usually with a crimped hair

called wool and often with horns forming a lateral spiral. Sheep

are herbivorous mammals. Ram is a male sheep while a female sheep is called a ewe.

Sheep are descended from the wild mouflon of Europe and Asia. The average life

expectancy of a sheep is 10 to 12 years, though some sheep may live as long as 20 years.

In the United States, over one third of sheep deaths in 2004 were caused by predation.

Birth rate for sheep is one to three times in a year and they can breed 3 lambs once.

Sheep have good hearing, and are sensitive to noise. More than that, sheep can see

behind themselves without turning their heads because it is have horizontal slit-shaped

pupils with visual fields of approximately 270 to 320. Sheep have little ability to defend

themselves, compared with other species. Even if sheep survive an attack, they may die

from their injuries or simply from panic because sheep are timid, nervous and easily

frightened animals and for the most part defenseless against predators like coyotes and

wild dogs. Usually, sheep will flock together in large numbers in order to survive and to

run away from predators. In addition, sheep do not like to walk in water or move through

narrow openings. They prefer to move into the wind and uphill than down wind and

downhill. (Wanda Embar (2007)
http://en.wikipedia.org/wiki/Crimp_(wool)http://en.wikipedia.org/wiki/Laterallyhttp://en.wikipedia.org/wiki/Spiralhttp://en.wikipedia.org/wiki/Herbivoroushttp://en.wikipedia.org/wiki/Mouflonhttp://en.wikipedia.org/wiki/Mouflonhttp://en.wikipedia.org/wiki/Herbivoroushttp://en.wikipedia.org/wiki/Spiralhttp://en.wikipedia.org/wiki/Laterallyhttp://en.wikipedia.org/wiki/Crimp_(wool)


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3.0 OBJECTIVE

a. To help the shepherd to make sure that the sheep will not get extinct.

b. To investigate the effect of sheep in the presence of coyote.

c. To determine the nature of stability of population of sheep and coyote.

d. To predict the population of sheep and coyote in 10 years.

4.0 LITERATURE REVIEW

The research of predator and prey also had been done at University of Colorado

Boulder in 2012. He states that the Lotka-Volterra model of prey and predator interaction

is essentially a kinetic model. He also uses sheep as a prey while coyote is the predator.

His basic Lotka Volterra model use time in day unit. Polymath program is used in order

to determine the nature of stability of his model. The graph of sheep and coyote

population versus time is periodic and isolate. The result shows that when he decreases

the initial population of coyote from the basic model, there is a larger isolation of sheep

population. It means that when the population coyote is decrease, the population of sheep

will increase. Hence, there is inversely proportional relation between sheep and coyote

population.Learn Cheme (2012).

Takeuchi Y. et al. (2005), have done a research with consider the evolution of a

system composed of two predatorprey deterministic systems described by Lotka

Volterra equations in random environment. They are proved that by using the influence of

telegraph noise, all positive trajectories always go out from any compact set of intR2+

with probability one if two rest points of the two systems do not coincide. They found

that in case where they have the rest point in common, the trajectory either converges to

the rest point or leaves from any compact set of intR2+. The system is neither permanent

nor dissipative if there is escape of the trajectories from any compact set.


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5.0 ANALYSIS

Beals M. et al. (1999) stated that the Lotka-Volterra model is composed of a pair

of differential equations that describe predator-prey dynamics in their simplest case

consist of one predator population and one prey population. It was developed

independently by Alfred Lotka and Vito Volterra in the 1920's, and is characterized by

oscillations in the population size of both predator and prey, with the peak of the

predator's oscillation lagging slightly behind the peak of the prey's oscillation.

Predators and prey can influence one another's evolution. There are several simplifying

assumptions for this model:

(a)The prey population will grow exponentially when the predator is absent

(b) The predator population will starve in the absence of the prey population (as

opposed to switching to another type of prey)

(c)Predators can consume infinite quantities of prey

(d)There is no environmental complexity (in other words, both populations are

moving randomly through a homogeneous environment).

Our variable is x, the number of sheep at time t, and y, the number of coyote at

time t. We assume that t is in a day. First assumption, we assume that when the coyote is

absent, the sheep population will increase their numbers about 2.19% by calculating their

growth rate(8/year). We model the growth of the sheep by the differential equation.

when y=0

Then, we assume that in the absence of the sheep, the coyote will starve and lead

to death. The rate of cheetah population is declining about 0.03% by calculating the rate

of their life expectancy. Life span for a coyote is around 10 years(1/3650). Hence, the

differential equation will be

when x=0


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If the sheep and coyote at the same area it is probably that the rate of sheep killed

by the coyote is proportional to the number of sheep and proportional to the number

coyote. If there are plenty of sheep, the population of the coyote will increase rapidly

because they get enough food. The differential equation for sheep is

A research found that the proportion of sheep population which killed by one

coyote at one time is 0.00012. Similarly, the differential equation for coyote is

The constant 0.0000024 (0.00012*0.02) represent the proportion of newborn

coyote in coyote population due to food value of one sheep multiply with proportion of

sheep population which killed by one coyote at one time


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6.0 RESULTS AND DISCUSSION6.1 Basic Model

x = 0.0219x (0.00012) xy

y= -0.0003y + (0.0000024) xy

x = 2000, y = 50

At t=0, the initial population of sheep is 2000 and the initial of coyote is

50. The growth rate of sheep is increase and also the growth rate of coyote is

increase because they have a lot of source of food. When the population of coyote

increases at t= 200days, the population of sheep will decrease and vice versa. This

happen periodically every 7000days.In this case, we assume both species does not

have any immigration or emigration. This is called basic Lotka Volterra model.

As we want to see clearer the behavior of this model, through phase portrait, we

can know the behavior of the model shown below.


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The phase portrait shows that there are two criticals point in this basic

model. First critical point is (125,183) and it shows that the behavior of the model

is stable center. This behavior shows that the periodically in the system like in the

explanation of graph above. This shows that the coexistence for both species. If

the initial number of populations if sufficiently close to the equilibrium point then

the interaction of sheep and coyote will remain at 125 and the populations of

coyote at 183. The second points is unstable saddle point at (0, 0) where its show

no life at all at this equilibrium point. The behavior of the phase diagram also

shows that the trajectories moving towards equilibrium and then depart from it. If

the initial value is close to (0, 0) equilibrium point, then this equilibrium

correspond to a state of coexistence where theoretically, the population of both of

sheep and coyote remain extinct forever.


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6.2 Modify value of initial populations

x = 3500, y = 250

x = 0.0219x (0.00012) xy

y= -0.0003y + (0.0000024)xy

The initial population is modifying for sheep is 3500 while for coyote is

250. As we can see from the graph above, at t=0, the population of sheep is higher

than coyote. Since sheep is the main source of food for coyote, when the sheep

population is increased, the population of coyote also increases because the

growth of rate for coyote also increases. This is due to the sufficient of food

resources. Since the population of predator increase, the population of sheep

decrease and become increase again at t= 6000days .this situation is periodically

happen in this system. Compare to the previous when initial population for sheep

is 2500 and the population of the coyote is 50, the sheep population will increase

at 7000 days which is 1000days later than modify initial model. This proves that

the initial population of the model may affect the time of species to increase or

decrease. It also important because we can know either that species will go extinct

or coexistence based on the initial population.


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6.3 One of the species is affected

i) Sheep is affecteda) Assuming that constant number of population is depleted

x= 0.0219x 0.00012 xy- 0.06

y= - 0.0003y + 0.0000024 xy

We assume that if the shepherd sold his sheep 25 per year, then

= 0.06 per day.

As we can see, the initial of sheep is 2000 and 50 of coyote but when

sheep is sold 25 per year, the population of sheep is decreased and the population

of coyote also decreased. This is happen because the source of food for coyote isstart to depleted then coyote do not have source of food and it will affected the

growth rate of coyote. The sheep population is start to decrease and still remain at

the population as the time goes.


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As we can see, there coexistence of both species at critical point

(125,178), the trajectories of direction field is moving toward to the equilibrium

point. This explains that sheep will coexist at 125 and coyote will coexist at 178

and the population size remains at this level indefinitely. The other point is at

critical point (3, 0) and this is saddle point. The trajectories moving toward

critical point and then depart from it. This explains if the initial number of sheep

and coyote are sufficiently close to (3,0), then the interaction process will

ultimately lead to survival of sheep at 3 and the extinction of coyote.


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b) Assuming that constant proportion of the population is depletedx= 0.0219x 0.00012 xy- 0.001x

y= - 0.0003y + 0.0000024 xy

In this case we assume 0.1% of the sheep population will depleted at that

time due to the rate of selling of sheep.

In this case, during t=0, the initial population of sheep is 2000 and the

population of coyote is 50. At the first, the population of sheep is increased and

the population of coyote also increases because coyote got more food. This will

lead to coyotes rate of birth increase. But when the population of sheep increases,

the sheep population will decrease. As the sheep population is decrease, source of

food for coyote is also decrease then population of coyote also decrease. Due to

the decrease of coyote population, the sheep population will increase. This phase

was happening periodically in 7000 days interval.


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From the above phase portrait, we have two critical points. First is at

(125,174) is a stable center. Thus it is stable. All trajectories approach around the

equilibrium. This equilibrium point corresponds to a state periodic and

coexistence of both species. Theoretically, the population of both sheep and

coyote remain at (125,174) forever. The second critical point (0, 0) is a saddle

point. Thus it is unstable. All trajectories nearby approach this equilibrium and

then depart from it. This equilibrium point correspond to a state of coexistence

where the population of sheep and coyote will remain at (0, 0) if the initial value

closed to this equilibrium point.


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ii) Coyote is affecteda) Assuming that constant number of population is depleted

x= 0.0219x0.00012 xy

y= - 0.0003y + 0.0000024 xy0.0274

We assume that 10 coyote is haunted per year.

= 0.0274

At initial, the population of sheep is 2000 and the population of coyote is

50. From the first 500 days, we can see that since the population of sheep is

higher. The population of coyote also start to increase at t= 125 days. At t =

250days, the population of coyote is 500 and at that time the population of sheep

also decrease. The source of food for coyote is decrease so the population of

coyote also decreases.


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From the above phase portrait we have one critical point. This critical

point (188, a83) is an unstable period. All trajectories will move from the center

of critical point and move to outside. This equilibrium point corresponds to a state

of coexistence for both species. The population size remains at (188,183) level

indefinitely. The population size for sheep will remain at 188 and the coyote

population will remain at 183.


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b) Assuming that constant proportion of the population is depletedx= 0.0219x 0.00012 xy

y= - 0.0003y + 0.0000024 xy0.001y

We are assume that 0.001 rate of death of coyote per year.

At t=0, the population of sheep is 2000 and the population of coyote is 50.

The population of sheep increase up to 7800 and at t=125 days, the population of

coyote also start increase. This is due to the increase of source of food, so the

coyotes rate ofgrowth rate will also increase. At t= 250 days, the population of

coyote is 500 but the population of sheep also decrease. As the population of

sheep decrease the population of coyote also decrease because lack of food

source. At t=2000 days, the population of sheep become increase again and this is

periodically every 2000 days


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From phase potrait, we have two criticals point. First is (542,183) is an

unstable center. All trajectories periodic and its shows that the coexistence of both

species. The sheep is 542 and the coyote is 183 . the population size remains at

this level forever. The second critical point is (0, 0) is unstable saddle point. All

trajectories nearby approach this equilibrium and then depart from it. This

equilibrium point correspond to a state of coexistence where theoretically, the

population of both sheep and coyote remain at (0, 0) forever if the initial its closed

to this critical point.


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6.4 Both of Sheep and Coyote are Affected

i) Sheep and coyote affected by constant number.

x= 0.0219x 0.00012 xy 0.06

y= - 0.0003y + 0.0000024 xy 0.0274

The model for this graph are affected when 25 sheep are going to be

sell each year while the 10 of coyotes die each year due to hunt . The graph

above shows the population of sheep increases and decreases drastically from

day zero to day 250 but decrease slowly after that. The population of coyote

also increases slowly until day 250 but then decrease slowly. The populations

of these two species are look to be constant for a long period which means

they are coexist for a certain time period. Although, the populations are not

look like to be constant or coexist after day 3250. They are look to be extinct.

This is proven by the nature stability of its critical point or equilibrium point.At point (188, 180) all trajectories in the neighborhood of the fixed point

spiral away from the fixed point with ever increasing radius. This show the

system is unstable and behaves as an unstable oscillator.


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The nature of this stability can be visualized as a vector tracing a

spiral away from the fixed point (a). The plot of response with time of this

situation would look sinusoidal with ever-increasing amplitude (b), as shown

below.

(a)


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(b)

This situation is undesirable when attempting to control the

population of sheep and coyote. This is because if there is a change in the

process, arising from the process itself or from an external disturbance, the

system itself will not go back to steady state which means it show that astime goes the population will extinct. By looking to this result, we actually

can help the shepherd to estimate their population of sheep each time. The

model can help them to know what they should do or when they should put

additional sheep to the cage to make the population of sheep remain exist and

their business will not end.


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ii) Sheep and coyote affected by constant proportion.

x= 0.0219x 0.00012 xy 0.06

y= - 0.0003y + 0.0000024 xy 0.0274

The model for this graph are affected when assuming around 0.1

percent of sheep and coyotes population are going to be die each time. The

graph above shows the population of these two species are periodic and the

pattern periodic every 2000 days. This graph is look like the basic graph but

the different is the pattern periodic for basic model is every 6000 to 7000

days and the maximum number that sheep can achieve is only 3500 compare

than this model where the growth rate is higher because in every 2000 days

the population of sheep can achieve 7050 sheep. For the modified one, it is

better than the basic model because it has been considers the normal dies for

the two species and period to the population recover is lesser with the higher

maximum population.


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Based on the graph, when the population of sheep is high, the

population of coyote will start increase but when the population of coyotes

keeps increase the population of sheep will decrease. Then, the two

populations will decrease. We can see for every period we have 3 phases,

which are;

1) 1st

phase is from day zero to day 200

a) Population of sheep and coyote increase

i) Population of sheep increase with normal birth rate

ii) Population of coyote increase because have enough food when it

interact with sheep as it food.

2) 2nd

phase from day 200 to day 250

a) Population of sheep decrease but population of coyote increase

i) Population of sheep decreases because it has been eaten by

coyote and dies normally.

ii) Population of coyote increase because it still has food to be eaten

and the growth is still not affected by the decreases of sheep

population.

3) 3rd

phase from day 250 to day 2000

a) Sheep and coyote decrease

i) Population of sheep decreases because it becomes the prey to

coyote which are increases in the 2nd

phase.

ii) Population of coyote decreases because there is lack of food

resources due to the drastically decreasing of sheep.

Since the graph is periodic, we know that the population of these two

species will coexistent. This is proven with the nature of stability by looking

to the phase portrait which show the trajectories of an equilibrium point

(542,174). At this point, the trajectories will move around the critical point.

This portrait shows the population of these two species will remain exist if

there are no other significant forces by nature or by plan. The system behaves

as a not damped oscillator.


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This nature of stability can be visualized in two dimensions as a

vector tracing a circle around a point (a). The plot of response with timewould look sinusoidal (b). The figures as shown below;

(a)


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(b)

This situation is common in many control system arising out of

competing controllers and other factors. Even so, this is usually undesirable

and considered as an unstable process since the system will not go back tosteady state following a disturbance. This is because; the population is not

remaining but always changes in the period of time. By looking to this result,we also can help the shepherd to estimate their population of sheep eachtime. They can know when the business can give them a better return and

when they will get nothing due to the population of the sheep.


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7.0 CONCLUSION

In the present work, we have been determining the phase potrait and the nature of

stability of each model that we have modified. The solution solved by using pplan8 and

odesolve of Math lab. The nature of stability of critical point for each model has been

achieved by using Jacobian matrix equation. From the findings, it is concluded that there

is coexistence between the populations of two species. In this case, the species of coyote

and sheep. In addition, both populations of the two species always changes in the period

of time.


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8.0 REFERENCES

Beals M. et al. (1999). Predator-Prey Dynamics: Lotka-Volterra. Retrieved June

15, 2013, from

http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html

Learn Cheme (2012). Predator - prey Model Kinetics. University of Colorado

Boulder. Retrived June 14, 2013, from

http://www.youtube.com/watch?v=7o5XweHevbQ

National Geographic Society (2013). Coyote. Retrieved June 14, 2013, from

National Geographic Site:http://animals.nationalgeographic.com

Ncwildlife (2012). Fox and Coyote Populations Study. Retrieved June 15, 2013, from

Ncwildlife site:

http://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_Coyot

ePopulationsReport.pdf

Takeuchi Y. et al. (2005). Evolution of predatorprey systems described by a Lotka

Volterra equation under random environment. Journal of Mathematical

Analysis and Applications. J. Math. Anal. Appl. 323, 938957.

Wanda Embar (2007). Sheep. Retrieved June 15, 2013, from veganpeace site:

http://www.veganpeace.com/animal_facts/Sheep.htm

Wikipedia (2013). Sheep. Retrieved June 15, 2013, from Wikipedia site:

http://en.wikipedia.org/wiki/Domestic_sheep
http://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.htmlhttp://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.htmlhttp://www.youtube.com/watch?v=7o5XweHevbQhttp://www.youtube.com/watch?v=7o5XweHevbQhttp://animals.nationalgeographic.com/animals/mammals/coyote/?source=A-to-Zhttp://animals.nationalgeographic.com/animals/mammals/coyote/?source=A-to-Zhttp://animals.nationalgeographic.com/animals/mammals/coyote/?source=A-to-Zhttp://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_CoyotePopulationsReport.pdfhttp://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_CoyotePopulationsReport.pdfhttp://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_CoyotePopulationsReport.pdfhttp://www.veganpeace.com/animal_facts/Sheep.htmhttp://www.veganpeace.com/animal_facts/Sheep.htmhttp://en.wikipedia.org/wiki/Domestic_sheep#Senseshttp://en.wikipedia.org/wiki/Domestic_sheep#Senseshttp://en.wikipedia.org/wiki/Domestic_sheep#Senseshttp://www.veganpeace.com/animal_facts/Sheep.htmhttp://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_CoyotePopulationsReport.pdfhttp://www.ncwildlife.org/Portals/0/Learning/documents/Species/Fox_CoyotePopulationsReport.pdfhttp://animals.nationalgeographic.com/animals/mammals/coyote/?source=A-to-Zhttp://www.youtube.com/watch?v=7o5XweHevbQhttp://www.tiem.utk.edu/~gross/bioed/bealsmodules/predator-prey.html


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9.0 APPENDICS

Let the nonlinear autonomous system be

And () is the critical point.

Jacobian matrix :

)

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