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

Sorting, Sorting The Sorting is a process of putting a list in order; ...

Sorting The Sorting is a process of putting a list in order; either in descending (maximum to minimum), or ascending (minimum to maximum) order. For illustration, here is a li

Rational function, Give the formula for a rational function that has a hole...

Give the formula for a rational function that has a hole at x=7 & vertical asmptote at x=-3/2

Creating customized strings, Creating Customized strings: There are ma...

Creating Customized strings: There are many built-in functions which create customized strings, including blanks, char, and sprintf. We know that the char function can be u

Plot types - plot functions, Plot types: Besides plot and bar, there a...

Plot types: Besides plot and bar, there are another plot types like stem plots, histograms, stem plots, area plots and pie charts, and also other functions which customize the

Built-in functions for complex numbers, Built-in functions for Complex numb...

Built-in functions for Complex numbers:   We know that in MATLAB both i and j are built-in functions which return √-1 (therefore, they can be thought of as built-in constants).

Pie chart - plot functions, Pie chart - plot functions: The MATLAB als...

Pie chart - plot functions: The MATLAB also has a function pie which will generate a pie chart. Calling the function with the form pie (vec) draws a pie chart, by using the pe

Example of logical built-in functions, Example of Logical built-in function...

Example of Logical built-in functions: For equivalent to all the function, we should make sure that the entire elements in the vector are logically true. The one way of doing

Writing variables to a file, Writing variables to a File: The save com...

Writing variables to a File: The save command is used to write variables to a file, or to append the variables to a MAT-file. By the default, save function writes to a MAT-fil

Illustration of spreadsheet files, Illustration of Spreadsheet Files: ...

Illustration of Spreadsheet Files: This reads the numbers in a double vector variable nums and the text in a cell array txt (the xlsread function forever returns the numbers f

Dimensions - matrix, Dimensions - matrix: The size and length function...

Dimensions - matrix: The size and length functions in the MATLAB are used to find array dimensions. Length function returns the number of elements in the vector. The size func

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