Predator-Prey Relationships By: Maria Casillas, Devin Morris, John Paul Phillips, Elly Sarabi, & Nernie Tam

Formulation of the Scientific Problem n There are many instances in nature where one species of animal feeds on another species of animal, which in turn feeds on other things. The first species is called the predator and the second is called the prey. n Theoretically, the predator can destroy all the prey so that the latter become extinct. However, if this happens the predator will also become extinct since, as we assume, it depends on the prey for its existence.

n What actually happens in nature is that a cycle develops where at some time the prey may be abundant and the predators few. Because of the abundance of prey, the predator population grows and reduces the population of prey. This results in a reduction of predators and consequent increase of prey and the cycle continues. -Predator -Prey

An important problem of ecology, the science which studies the interrelationships of organisms and their environment, is to investigate the question of coexistence of the two species. To this end, it is natural to seek a mathematical formulation of this predator-prey problem and to use it to forecast the behavior of populations of various species at different times.

Risk and Food Availabilty n Sharks appear to be a major threat to fish n Availabilty of prey helps animals decide where to live

Predator-Prey Model: Fish & Sharks n We will create a mathmatical model which describes the relationship between predator and prey in the ocean. Where the predators are sharks and the prey are fish. n In order for this model to work we must first make a few assumptions.

Assumptions n 1. Fish only die by being eaten by Sharks, and of natural causes. n 2. Sharks only die from natural causes. n 3. The interaction between Sharks and Fish can be described by a function.

Differential Equations and how it Relates to Predator-Prey One of the most interesting applications of systems of differential equations is the predator-prey problem. In this project we will consider an environment containing two related populations-a prey population, such as fish, and a predator population, such as sharks. Clearly, it is reasonable to expect that the two populations react in such a way as to influence each other’s size. The differential equations are very much helpful in many areas of science. But most of interesting real life problems involve more than one unknown function. Therefore, the use of system of differential equations is very useful. Without loss of generality, we will concentrate on systems of two differential equations.

Vito Volterra n Born –Ancona, Papal States (now Italy) –May 3rd, 1860

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n Age 2: –Father passed away, family was poor. n Age 11: –Began studying Legendre’s Geometry n Age 13: –Began studying Three Body Problem –Progress!

n 1878: –Studied under Betti in Pisa n 1882: –graduated Doctor of Physics –thesis was on hydrodynamics

n 1883: –Professor of Mechanics at Pisa –Chair of Mathematical Physics –conceived idea of a theory of functions

n 1890: –Extended theory of Hamilton and Jacobi n : –published papers on partial differential equations

n 1896: –published papers on integral equations of the Volterra type. n WWI: –Air Force –Scientific collaboration

n Post War: –University of Rome –Verhulst equation –logistic curve –predator-prey equations!

n 1922: –Italian Parliament n 1930: –Parliament abolished n 1931: –forced to leave University of Rome

Predator-Prey Populations n 1926: –published deduction of the nonlinear differential equation –similar to Lotka’s logistic growth equation –Lotka-Volterra Equation

Crater Volterra n Lunar Crater –Location n Latitude: 56.8 degrees North n Longitude: degrees East –Diameter n 52.0 kilometers!

n 1938: –Offered degree by University of St Andrews!

n Died –Rome, Italy –October 11th, 1940

Alfred James Lotka n Born: –1880 –Lviv (Lemberg), Austria (Ukraine)

n Chemist n Demographer n Ecologist n Mathematician

n 1902: –Moved to the United States n Chemical Oscillations n 1925: –Wrote Analytical Theory of Biological Populations

n Predator-Prey model –independent from Volterra –Analysis of population dynamics n Metropolitan Life

Population Assoc. of America n Nonprofit organization n Scientific organization n Promoting improvement of human race n Membership now 3,000! n Annual meetings

n Famous for Avant La Lettre n Power Law –(C/(n^{a})) –Where C is a constant –If a=2, then C=(6/(pi^{2}))=0.61

Alfred James Lotka n Died: –1949 –USA

The Lotka-Volterra Model n System n F'(t)=aF-bF²-cFS n S'(t)=-kS+dSF n Initial Conditions n F(0)=F 0 n S(0)=S 0 F(t) represents the population of the fish at time t S(t) represents the population of the sharks at time t F 0 is the initial size of the fish population S 0 is the initial size of the shark population

Understanding the Model n F'(t)=aF-bF²-cFS n F’(t) the growth rate of the fish population, is influenced, according to the first differential equation, by three different terms. n It is positively influenced by the current fish population size, as shown by the term aF, where a is a constant, non-negative real number and aF is the birthrate of the fish. n It is negatively influenced by the natural death rate of the fish, as shown by the term -bF², where b is a constant, non-negative real number and bF² is the natural death rate of the fish n It is also negatively influenced by the death rate of the fish due to consumption by sharks as shown by the term -cFS, where c is a constant non-negative real number and cFS is the death rate of the fish due to consumption by sharks.

n S'(t)=-kS+dSF n S’(t), the growth rate of the Shark population, is influenced, according to the second differential equation, by two different terms. n It is negatively influenced by the current shark population size as shown by the term -kS, where k is a constant non-negative real number and S is the shark population. n It is positively influenced by the shark-fish interactions as shown by the term dSF, where d is a constant non-negative real number, S is the shark population and F is the fish population.

Equilibrium Points n Once the initial equations are understood, the next step is to find the equilibrium points. n These equilibrium points represent points on the graph of the function which are significant. n These are shown by the following computations.

n Let X=(dF/dt)=F(a-bF-cS) n Let Y= (dS/dt)=S(-k+dF) n To compute the equilibrium points we solve (dF/dt)=0 and (dS/dt)=0 n (dF/dt)=0 when F=0 or a-bF-cS=0 n solution: F=(a-cS)/b

dS/dt=0 when S=0 or -k+dF=0 Solution:{F=(k/d)} Now we find all the combinations: One of our equilibrium points is (0,0). For F=(a-cS)/b: When S=0, then F=((a-c(0))/b)= (a/b) Thus, one of our equilibrium points is ((a/b),0). For F=((a-cS)/b) and F=(k/d): (k/d)=((a-cS)/b), Solution is: {S=((-kb+ad)/(dc))} Thus, one of our equilibrium points is ((k/d),((-kb+ad)/(dc))). Our equilibrium points are (0,0), ((a/b),0), and ((k/d),((-kb+ad)/(dc))).

Now, to study the stability of the equilibrium points we first need to find the Jacobian matrix which is: J(F,S)= = To study the stability of (0,0): J(0,0)=det = (a- λ )(-k- λ ), Solution is: { λ =a},{ λ =-k} semi-stable since one eigenvalue is negative and one is positive.

To study the stability of ((a/b),0): J((a/b),0)=det = (-a- λ )(-k+a(d/b)- λ ), Solution is: { λ =-a},{ λ =((-kb+ad)/b)} stable if λ =((-kb+ad)/b) < 0 (i.e. ad < kb) semi-stable if λ =((-kb+ad)/b) > 0 (i.e. ad > kb) To study the stability of ((k/d),((ad-kb)/(cd))): J((k/d),((ad-kb)/(cd)))=det =det = (( λ kb+ λ ²d-k²b+kad)/d)

Solution is: { λ =(1/(2d))(-kb+(k²b²+4dk²b-4kad²)^1/2)} { λ =(1/(2d))(-kb-(k²b²+4dk²b-4kad²)^1/2)} If we simplify a little more, we get: λ =(1/(2d))(-kb-(k²b²+4dk²b-4kad²)^1/2) = -(1/2)((kb+i(k)^1/2(-kb²-4dkb+4ad²)^1/2)/d) λ =(1/(2d))(-kb+(k²b²+4dk²b-4kad²)^1/2) = -(1/2)((kb-i(k)^1/2(-kb²-4dkb+4ad²)^1/2)/d) Stable since both of the real parts are negative. The imaginary numbers tells us that it will be periodic.

Case 1 (a λ >bk) u(x,y)=x(6-2x-4y) v(x,y)=y(-3+5x) x(0)=1 y(0)=.5 fish sharks

x(0)=2 y(0)=3 sharks fish

x(0)=.5 y(0)=1.5 sharks fish

x(0)=.5 y(0)=.5 sharks fish

Case 2: (a λ <bk) u(x,y)=x(2-6x-4y) v(x,y)=y(-3+5x) x(0)=1 y(0)=.5 sharks fish

x(0)=2 y(0)=3 sharks fish

x(0)=0.5 sharks fish

x(0)=.5 y(0)=.5 sharks fish

X(0)=.1 y(0)=.1 sharks fish

Case 3: All constants are equal u(x,y)=x(1-1x-1y) v(x,y)=y(-1+1x) sharks fish

X(0)=2 y(0)=3 sharks fish

X(0)=.5 y(0)=1.5 sharks fish

X(0)=.5 y(0)=.5 sharks fish

Case #4: (b=0) u(x,y)=x(2-0x-1y) v(x,y)=y(-1+1x) x(0)=1 y(0)=0.5 sharks fish

x(0)=2 y(0)=3 sharks fish

x(0)=0.5 y(0)=1.5 sharks fish

x(0)=0.5 y(0)=0.5 sharks fish

Case 5: ((k/ λ )=((a λ -bk)/(c λ ))) u(x,y)=x(2-1x-1y) v(x,y)=y(-1+1x) x(0)=1 y(0)=0.5 sharks fish

x(0)=2 y(0)=3 sharks fish

x(0)=0.5 y(0)=1.5 sharks fish

x(0)=0.5 y(0)=0.5 sharks fish

With the eigenvalues -bk+i(4akk -4bk -bbkk)^1/2 and -bk-i(4akk -4bk -bbkk)^1/2, we are able to calculate the period of the oscillations: (2  )/(4akk -4bk -bbkk)^1/2 This is the rough length of one oscillating cycle for this model.

The eigenvalues also allow us to describe the cyclic variation of this model by using their properties in developing U(t) and V(t): U(t) = (e^t(-bk/2 ))(kK/ )cos(t ((4akk -4bk -bbkk)^1/2)+  ) V(t) = (e^t(-bk/2 ))(kK/c )((a -bk)^1/2)*sin(t ((4akk -4bk - bbkk)^1/2)+  ) And by substituting, we get: F(t) = k/ (1+ (e^t(-bk/2 ))Kcos[t {(4akk -4bk -bbkk)^1/2}+  ] S(t) = {(a -bk)/c }*(1+ {e^t(-bk/2 )}*[{k/(a -bk)}^1/2]*Ksin[t {(4akk -4bk -bbkk)^1/2}+  ])

From those equations, we are able to get the amplitudes of the oscillations, which are: For F(t); K(k/ ){e^t(-bk/2 )} And for S(t); K(k/c ){(a -bk)^1/2}{e^t(-bk/2 )} With K and  representing the initial conditions {F(0), S(0)} And the average number of F(t) is: k/ and S(t)’s average number is: (a -bk)/c Those numbers are identical to the coordinates of the critical point.

Both the exponential and trigometrical aspect of the solutions of F(t) and S(t) tells us that the graph of the equations will show an infinite spiraling pattern towards the critical point for the first case. The first case has a/b greater than k/, where a/b is the stable point for the fish population in a shark-free world, and k/, of course is the critical point for the fish population living with sharks. This case holds true regardless of the initial conditions, as long as F(0)>0, and S(0)>0.

For the second case, when a/b is less than k/, we arrive at an interesting conclusion, which is supported by simple algebra. When a/b < k/, then for the shark equation, the critical point becomes a negative number! a/b a < bk so S(t) = (a -bk)/c results in a negative number. Therefore in the second case, the shark population will die out REGARDLESS of the initial conditions! So the solution would converge to the shark-free stable point.

For the third case, what if all of the constants were the same? A simple glance at the equations tells us that this would be similar to the second case: we get a/b=k/ =1, yet the critical point would be (1,0) which is on the y-axis (S) and identical to the stable point for the fish population in a shark-free world. So here the sharks die out again. (But it’s hard to feel sorry for sharks!)

For the fourth case, we make b=0 which turns the model into the simplest form of the predator/prey model. The new equations look like this: F’ = F(a-0F-cS) = F(a-cS) S’ = S(-k+ F) So the critical point becomes (k/, a/c) and we get an ellipse around the critical point, the shape and size depending on the constants and initial conditions. So both the fish and shark populations wax and wane in a cyclic pattern with the sharks lagging behind the fish.

Now for the fifth case, we pose the question: What happens when F(t) = S(t), i.e. k/ = (a -bk)/c ? Answer: This is pretty much similar to the first case, since a/b > k/, with a simpler spiral as the result. The only significant impact is the location of the critical point.

There are other cases that we have yet to explore here, such as a=0, c=0, k=0, =0, or a combination of those, but those would render the model meaningless, as they would cancel the relationship between the fish and the sharks or eliminate the fish’s growth rate or the shark’s death rate.

In Conclusion, This Lotka-Volterra Predator-Prey Model is a rudimentary model of the complex ecology of this world. It assumes just one prey for the predator, and vice versa. It also assumes no outside influences like disease, changing conditions, pollution, and so on. However, the model can be expanded to include other variables, and we have Lotka-Volterra Competition Model, which models two competing species and the resources that they need to survive. We can polish the equations by adding more variables and get a better picture of the ecology. But with more variables, the model becomes more complex and would require more brains or computer resources.

This model is an excellent tool to teach the principles involved in ecology, and to show some rather counter-initiative results. It also shows a special relationship between biology and mathematics. Now, what does this has to do with orbital mechanics? Simple: this model is similar to the models of orbits with those spirals, contours and curves. We can apply this model with constants representing gravitational pulls and speeds of bodies.

Conclusion n Hopefully, you now have a little insight into the thinking that was behind the creation of the Lotka-Volterra model for predator- prey interaction! Thank you, and this has been a fun project!

Work Cited n Boyce, William. Elementary Differential Equations. New York: John Wiley & Sons, Inc., 1986 n Cullen, Michael & Zill, Dennis. Differential Equations with Boundary-Value Problems. Boston: PWS-Kent Publishing Company, 1993 n Zill, Dennis. A First Course in Differential Equations: The Classic Fifth Edition. California: Brooks/Cole, 2001 n Neuhauser, Claudia. Calculus for Biology and Medicine: New Jersey, 2000 n Intoduction to the Predator Prey Problem. 8/20/02 n Mathematical Formulation. 8/20/02 n Lotka Volterra Model. 8/20/02 n Predator-Prey Modeling. rohan.sdsu.edu/~jmahaffy/courses/bridges/bridges00.htm; 8/22/02 n Predator Prey Model. 8/20/02