User Account

All Pages

Optimal foraging theory, MATLAB in Mathematics

Assignment Help:
This project requires you to use the ideas of Chapter 25 regarding maximization of a function. Here we assume that evolution has acted to generate highly efficient foragers. By highly efficient here we mean that the animals searching for food which are able to more rapidly obtain a high food intake rate (food eaten per unit time), will be more likely to survive and reproduce. Thus, if the characteristics that lead to high food intake efficiency (which may depend upon speed, visual or hearing skills, size, etc.) are heritable, then the characteristics which lead to higher efficiency will become more prevalent (increase in frequency) in the population. This area of science is called optimal foraging theory. For a readable description of this theory see David Stephens and John Krebs book Foraging Theory [60].
One of the main areas of foraging theory deals with animals which move from “patch” to “patch” across a landscape, depleting the food in each patch the longer they stay there. Patches could be individual plants with nectar in the plants flowers that is eaten by bees, or individual plants with seeds that are eaten by birds. One question in foraging theory is to deduce the rules for when an individual will leave one patch to search for a new patch. Numerous complications arise in this, but we will consider only a very simple situation, in which we have the following assumptions:
(1) There are no interactions with other foragers, so depletion of food in a patch is due only to the single forager under consideration.
(2) There is no randomness in the environment, so that each patch is identical in terms of how much resource it has before foraging starts.
(3) Once the patch is depleted it stays depleted. (4) There are many patches available. (5) The travel-time of the forager between patches is constant.
(6) The food available in the patch decreases exponentially as the forager spends time there.
467
468 Unit 6 Derivatives
Let t be the time spent in a patch before leaving, M be the time it takes to move between two patches, and K be the amount of food in a patch before any food has been eaten. Then the total amount of food eaten in a patch is
K(1 - e-ct) (27.1)
where c is the decay rate of available food in a patch once foraging starts. In Equation 27.1 e-ct represents the proportion of food left in the patch after the forager has eaten from the patch over time t. Thus, 1 - e-ct represents the proportion of food eaten from the patch over time t. When we multiply this term by K (the total amount of food in the patch before any is eaten) we obtain the amount of food eaten from the patch after the foraging is complete.
We define the one foraging bout as the time spent moving to the patch plus the time spent foraging in that patch. Thus, the total time of one foraging bout is
M + T. The food intake rate is the amount of food consumed divided by the time spent
consuming the food, i.e.
K ??1 - e-cT ??
(27.2)
T+M Using the example from Section 25.6 as an example, complete the following.
1. Think of Equation 27.2 as a function of the time spent in a patch before leaving, t. Take the first derivative of this function with respect to t. Notice, that when the derivative is set equal to zero, you cannot explicitly solve for t. Write an equation, in terms of t, c, and M that can be graphed and used to find the critical points.
2. Use Matlab to estimate the critical point of the derivative of Equation 27.2, using c = 4 and M = 10. Based on the graph you produced to find the value of the critical point correct to two decimal places. Explain why you know that a local maximum occurs at this point.
3. The critical point found in part 2 is the optimal time a forager should spend foraging in one particular path. Explain why it makes sense biologically that this point is a local maximum.
4. Repeat part 2 but use M = 1, 5, 15, and 20. Does the graph in each case indicate that the critical point is still a maximum? How do the optimal foraging times in these cases compare to the optimal foraging time you found in part 2.
5. Repeat part 2 but use c = 0, 2, 4, 6, 8, 10. Does the graph in each case indicate that the critical point is till a maximum? How to the optimal foraging times in these cases compare to the optimal foraging time you found in part 2.
Chapter 27 Unit Projects 469
6. In Matlab, create the following m-file.
OptimalForaging.m 1 M = [1 5 10 15 20];
2 c = [0 2 4 6 8 10]; 3z= 4 5
6 7
surface(c,M,z); xlabel(’c’); ylabel(’M’);
For z, construct a matrix where each column corresponds to an M value, (1, 5, 10, 15, 20), each row corresponds to a c value, (0, 2, 4, 6, 8, 10), and the value of each entry in the matrix is the corresponding optimal foraging time. For example, the optimal foraging time found in part 2 would be the entry in row 3, column 3. You already have several of the values from parts 2, 4, and 5. You will need to use Matlab to determine the rest of the values. The resulting matrix z should be a 6 × 5 matrix.
Once you have constructed matrix z, run the m-file OptimalForaging.m. A plot
will be generated. Use the button to rotate the graph to look at the surface plot from different angles. Describe what this 3-dimensional surface plot is showing you. What are the advantages and disadvantages of viewing a 3-d plot such as this?

Related Discussions:- Optimal foraging theory

Using functions with the vectors and matrices, Using Functions with the Vec...

Using Functions with the Vectors and Matrices: As MATLAB is written to work with the vectors and matrices, the whole vector or matrix can be passed as an argument to the funct

Counting in a while loop, Counting in a While Loop: Whenever it is not...

Counting in a While Loop: Whenever it is not known ahead of the time how many values will be entered into the script, it is often essential to count the number of values which

Changing dimensions, Changing Dimensions: In addition to transpose ope...

Changing Dimensions: In addition to transpose operator, the MATLAB has some built-in functions which change the dimensions or configuration of matrices, involving fliplr, resh

Strcat function - concatenation, Strcat function - Concatenation : The ...

Strcat function - Concatenation : The strcat function, though, will eliminate the trailing blanks from strings before concatenating. Note that in these illustrations, the trail

Indexed empty matrix, Indexed empty matrix: The Individual elements ca...

Indexed empty matrix: The Individual elements cannot be eliminated from matrices, as matrices always have the similar number of elements in every row. >> mat = [7 9 8; 4 6

Standard deviation, Standard Deviation The standard deviation is the sq...

Standard Deviation The standard deviation is the square root of variance: The built-in function in a MATLAB for the standard deviation is known as std; the standard dev

Types of user-defined functions, Types of User-defined Functions: We k...

Types of User-defined Functions: We know how to write a user-defined function, stored in an M-file, which computes and returns one value. This is merely one type of the functi

Illustration of standard deviation, Illustration of Standard Deviation ...

Illustration of Standard Deviation The less spread out the numbers are, therefore smaller the standard deviation will be, as it is a way of determining the spread of the data.

Example of symbolic expression, Example of Symbolic Expression When the...

Example of Symbolic Expression When there is more than one variable, the MATLAB selects which to solve for. In the illustration below, the equation ax 2 + bx = 0 is solved. Th

Program of built-in factorial function, Program of built-in factorial funct...

Program of built-in factorial function: Calling this function yields similar result as the built-in factorial function: >> fact(5) ans =   120 >> factorial(

Write Your Message!

Captcha
Order Assignment

Featured Services

Order Now

Popular Subjects

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

Submit Assignment

Academics

Major Subjects

Majors

Get In Touch

whatsapp: +91-977-207-8620

Phone: +91-977-207-8620

Email: [email protected]

TERMS & POLICIES

HELP & SUPPORT

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd