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MAC2282.991 – APPLIED CALCULUS PROJECT

INTERDEPENDENT MODELING: CUNICULTURE AND

AQUACULTURE

ADAM KOWALESKI, University of South Florida

Advisor:

DR. STANLEY C KRANC, Civil and Environmental Engineering, University of South

Florida

Spring Semester, 2014

Abstract: The body of work surrounding the use of rabbit excreta to fertilize tilapia

ponds strongly suggests a higher yield of fish when compared to traditional farming

methods. Interdependent models can be used by agricultural engineers to plan and

refine integrated farming systems. By adapting the Lotka-Volterra equations this

project seeks to model the production and harvest of rabbits and tilapia (a

commonly farmed meat fish) wherein the tilapia are dependent on rabbit excreta

for subsistence and reciprocally rabbits are fed from the proceeds of tilapia sold.

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TABLE OF CONTENTS

Problem Statement…………………………………………………………………..............…………3

Heun’s Method………………………………………………………………………..…………………………4

Solution Approach…………………….……………………………………………………………………….6

Discussion..……………………………………………………………………………………………………….7

Motivation……………………………………….……………………………………………………………….10

Recommendations…………………………………………………………………………………………..11

References…………………………………………….…………………………………………………………12

Nomenclature………………………………………………………………………………………………….12

Appendix………………………………………………………………………………………………………….13

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PROBLEM STATEMENT

The Lotka-Volterra equations are a popular method used in describing trends among animal populations; the most popular are known as predator-prey models.

These models, as the name implies, are effective in showing the dynamics of predators and prey in a given system. The goal of this project is to adapt the Lotka-

Volterra predator-prey model to show interdependence; particularly, an integrated-farming system using rabbit and tilapia wherein the tilapia (T) are completely dependent on rabbit (R) excrement for sustenance and the proceeds from tilapia

sold feed the rabbits. Both will be plotted against time (t).

We can make the following assumptions:

This is an infinite system with no space limitations Rabbits and tilapia harvest is proportional to the amount present

Money earned from tilapia production is used to purchase food for the rabbits

The Lotka-Volterra equations (Berryman, 1992, p. 1531) are as follows:

𝑑𝑅

𝑑𝑡 = 𝑎𝑅 − 𝛽𝑅𝑇

𝑑𝑇

𝑑𝑡 = −𝑐𝑇 + 𝛿𝑅𝑇

These are two coupled, nonlinear differential equations and subject to initial

conditions. a, c, β, and δ are all parameters set by the user, which ideally come from field data. A popular example of a predator-prey model demonstrates rabbits

(prey) and foxes (predators):

FIGURE 1

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We can see that the population of the two animals oscillates (this solution was completed using a method discussed in the next section). As the prey

population grows, so does that of the predator. Eventually the predators consume the prey too rapidly and both decline. This back and forth results in the oscillation.

The keen reader knows, though, that rabbits nor tilapia prey on each other; in fact, they are both vegetarians. So how will the Lotka-Volterra equation work

with two vegetarians? It won’t. Interdependence is our objective, so we will adapt the equation:

𝑑𝑅

𝑑𝑡 = −𝑎𝑅 + 𝛽𝑅𝑇

𝑑𝑇

𝑑𝑡 = −𝑐𝑇 + 𝛿𝑅𝑇

Notice that the signs of a, c, and β have been changed so that both equations are in the same form. The constants are as follows:

Constant Description

a Rabbit harvest rate

c Tilapia harvest rate

β Rabbit population growth rate

δ Tilapia population growth rate

With the signs of a and c negative both populations will be being harvested. By changing the sign of β it now matches that of δ and both populations will grow.

Further, they depend on each other – RT is present in both.

HEUN’S METHOD

We will compute our populations’ growth using Heun’s method, which is also

known as a predictor-corrector, or improved Euler method (Barker, 2002). It is important to note that we are not solving the differential equations; our goal is to

project values against a theoretical solution line (Barker, 2002). In other words we are looking for a numerical solution, not an analytical one.

Heun’s method has two steps. The first is Euler’s (or predictor) and the second is a correction step, Formulaically for Euler’s method we have (where t is

time and given):

𝑥𝑛+1 = 𝑥𝑛 + 𝛥𝑡

𝑦𝑛+1 = 𝑦𝑛 + 𝑓(𝑥𝑛, 𝑦𝑛) ∗ 𝛥𝑡

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Now for the corrector step, where Δx and Δy are the values from our first

step, the second is:

𝑥𝑛+1 = 𝑥𝑛 + 𝛥𝑡

𝑦𝑛+1 = 𝑦𝑛 + 𝑓(𝑥𝑛 + 𝛥𝑥, 𝑦𝑛 + 𝛥𝑦) ∗ 𝛥𝑡

Our final step is as follows:

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2(𝑠𝑡𝑒𝑝1 + 𝑠𝑡𝑒𝑝2) ∗ 𝛥𝑡

This project began by using Heun’s method to model simple problems like (where a is a constant):

𝑑𝑦

𝑑𝑥= 𝑎𝑦

Pictorially, we see the Euler, or predictor step, in Figure 1 as the left tangent prediction line. Then we see the corrector step as the prediction line based on slope

of right tangent line. When we take the average we find the middle orange line. This final step is significantly more accurate against the blue solution line.

FIGURE 2

An example of the predictor-corrector. From Heun’s Method, Theoretical

Introduction, by Barker, C. A. (2002). Retrieved from http://web.archive.org/web/20090212005921/http://calculuslab.deltacollege.edu/O

DE/7-C-2/7-C-2-h.html

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SOLUTION APPROACH

Since we understand the method, let’s establish our initial conditions. It

should be said that these are arbitrary beginning points which are not dictated by the process. In other words, Heun’s method will work with any positive initial conditions. We will work with R(0) and T(0) at t(0).

𝑅(0) = 10

𝑇(0) = 100

𝑡(0) = 0 𝑎𝑛𝑑 𝛥𝑡 = 1 (𝑚𝑜𝑛𝑡ℎ)

Let’s apply Heun’s method! Our adapted Lotka-Volterra equations again are:

𝑑𝑅

𝑑𝑡 = −𝑎𝑅 + 𝛽𝑅𝑇

𝑑𝑇

𝑑𝑡 = −𝑐𝑇 + 𝛿𝑅𝑇

𝐾𝑎𝑅 = −𝑎𝑅 + 𝛽𝑅𝑇

𝐾𝑏𝑅 = −𝑎(𝑅 + 𝐾𝑎𝑅) + 𝛽(𝑅 + 𝐾𝑎𝑅 ∗ 𝛥𝑡)(𝑇 + 𝐾𝑎𝑇 ∗ 𝛥𝑡)

𝐾𝑎𝑇 = −𝑐𝑇 + 𝛿𝑅𝑇

𝐾𝑏𝑇 = −𝑐(𝑇 + 𝐾𝑎𝑇) + 𝛿(𝑅 + 𝐾𝑎𝑅 ∗ 𝛥𝑡)(𝑇 + 𝐾𝑎𝑇 ∗ 𝛥𝑡)

To move one time step (from 0 to 1) we will now average the constants to approximate our next values:

𝑡(1) = 𝑡(0) + 𝛥𝑡

𝑅(1) = 1

2( 𝐾𝑎𝑅 + 𝐾𝑏𝑅) ∗ 𝛥𝑡

𝑇(1) = 1

2( 𝐾𝑎𝑇 + 𝐾𝑏𝑇) ∗ 𝛥𝑡

A convenient method to work in this space is by way of Excel. Here, the

formulas and initial conditions can be constructed and subsequently marched on to the desired time. For example:

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DISCUSSION

At the present time, we are without real-world data and will work with the following arbitrary constants:

Time Rabbits Tilapia kaR kbR kaT kbT

0 10 100 0.9 0.981 0 0.09

1 10.9405 100.045 0.985137323 1.0749661 0.09409232 0.192831574

2 11.9706 100.188 1.079605649 1.1795501 0.19742655 0.306192758

3 13.1001 100.44 1.184779277 1.296379 0.31137786 0.431711635

4 14.3407 100.812 1.302305808 1.4274159 0.43759473 0.5713519

5 15.7056 101.316 1.434174341 1.5750457 0.57806714 0.727499614

CASE a c β δ

1 0.01 0.01 0.001 0.001

2 0.05 0.5 0.001 0.0001

3 0.05 0.5 0.01 0.001

4 0.15 0.2 0.008 0.0008

5 0.1 0.2 0.001 0.02

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We can see multiple outcomes in these five cases. In CASE 1 we have applied the same growth and harvest rate. This produced a uniform growth.

In CASE 2 our harvest rate and growth for the tilapia are both

respectively less than that of the rabbit. This results in an abrupt population decrease in tilapia and a gradual decline to zero in the rabbits.

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CASE 3 is similar to CASE 2, however we have slightly increased our rabbit and tilapia growth rates. The outcome is the same, however we see a

short-lived production boost in the rabbits.

CASE 4 has higher growth constants than the prior two and adjust harvest rates. We are taking slightly less rabbits than tilapia. Starting with 100 tilapia helps the rabbit population peak at nearly 100 rabbits, but all at

the cost of the tilapia.

Lastly, we have reached equilibrium in CASE 5 we are constantly producing and harvesting the same amount for each animal.

All the examples here demonstrate the interdependence of the two

populations; this was the desired outcome of adapting the Lotka-Volterra equations. The most desirable outcome would be CASE 1, which has both

growth and profit.

MOTIVATION

The United Nations Population Fund estimates Earth’s population to increase

by one billion in the next twelve years (“Linking Population, Poverty and Development”, 2008). As communities expand, farm land decreases and more land-efficient farming methods will be necessary to meet demands. Agricultural engineers

are tasked with overcoming this obstacle.

Rabbits are an herbivorous animal that can be farmed in compact areas

because of their small size. They eat fibrous by-products, such as hay, which offers

no competition to other domestic animals like poultry and swine (Rukera Tabaro,

Mutanga, Rugege, & Micha, 2012).

Because of their small size, cages can be built in many configurations –

including over water. As an added benefit, farming rabbits above ground reduces exposure to parasites and bacterial infections caused by unsanitary conditions.

Rabbits breed year round, have a short gestation period at just 30 days, and offer sizeable, lean protein which can be taken at 8 weeks of age.

Tilapia are a fast-growing, vegetarian fish, which are commonly farmed for meat. Intensively fertilized ponds can produce between 9 and 50 kg/Ha/day of fish

meat (Franco, 1991). Traditionally tilapia are farmed in ponds, however like rabbits this condition can be simulated and space efficient.

Nutrients and energy contained in rabbit excreta (food droppings, feces, and urine) can be recycled by being converted into pond micro-organisms and algal

growth to be consumed by fish (Franco, 1991).

This system could be ideal in developing countries as it offers a high meat yield, low investment, land-use efficiency, and depending on scope, low labor. Setup can be achieved inexpensively and without high-tech equipment. Rabbits and tilapia

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are both easily handled and do not require any special processing. And, lastly, rabbits and tilapia are both diverse – there are a myriad of species available throughout the

world.

RECOMMENDATIONS

The scope of this presentation has worked with arbitrary values and limited

constants; there are seemingly endless additions. For example, we have not accounted for death in the animals. Mostly because we are harvesting both at a

specified rate, presumably before natural death. However, though, we could add a variable for natural death within the process. Another addition would be to make the harvesting rates a and c functions of time. It would be beneficial to harvest

larger quantities when the population begins to explode. Should the graph continue for CASE 1, we would see unrealistically high values. Adjusting a and c could to be

functions of time could address this.

The case examples presented here seem viable in theory. Actual experimentation is required to determine values for the constants and other possible variables to account for.

Lastly, this model is not exclusive to tilapia and rabbits. Other integrated

farming models exist. For example, fish produce ammonia which can be used as a plant fertilizer. Duck, poultry, and cow dung have also been used in fertilizing water to promote fish growth.

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REFERENCES

Barker, C. (2002). Numerical Methods for Solving Differential Equations, Heun’s

Method, Theoretical Introduction. Retrieved from

http://web.archive.org/web/20090212005921/http://calculuslab.deltacolleg

.edu/ODE/7-C-2/7-C-2-h.html

Berryman, A. A. (1992). The Origins and Theory of Predator-Prey Theory. Ecology, 73(4), 1530-1535. Retrieved from http://www.jstor.org/stable/1940005

Franco, L. (1991). Nile Tilapia production in tropical microcosms fertilized with

rabbit excreta (Unpublished thesis). Oregon State University, Oregon.

Retrieved from https://ir.library.oregonstate.edu/xmlui/handle/1957/37157

Rukera Tabaro, S., Mutanga, O., Rugege, D., & Micha, J. (2012). Rearing rabbits

over earthen fish ponds in Rwanda: effects on water and sediment quality,

growth, and production of Nile Tilapia Oreochromis niloticus. Journal Of

Applied Aquaculture, 24(2), 170-181. doi:10.1080/10454438.2012.679155

United Nations Population Fund. (2008) Linking Population, Poverty and

Development. Retrieved from https://www.unfpa.org/pds/trends.htm

NOMENCLATURE

Symbol Description Units

R Number of Rabbits Rabbits

T Number of Tilapia Tilapia

t Time Months

a Birth Rate of Rabbits Rabbits/Month

c Harvest Rate of Rabbits Rabbits/Month

β Birth Rate of Tilapia Tilapia/Month

δ Harvest Rate of Tilapia Tilapia/Month

k Parameter of Predictor Corrector Method

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APPENDIX

Figure 3: A small-scale, urban configuration using 35 gallon food-grade barrels and

an elevated rabbit cage.