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# Intro to Matlab MATH 410

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This 9 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 410 at College of William and Mary taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/231146/math-410-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Date Created: 10/29/15

Model 3a Allee effect Assumptions 1 If the population is too large then the growth is negative 2a If the population is too small then the growth is negative Example The fox squirrel a small mammal native to the Rocky Mountains If the population is too small fertile adults run the risk of not being able to find suitable mates so the growth is negative d szPlt1 gtlt5 lgt dt N M k 2 growth rate per capita coefficient N carrying capacity M sparsity constant and O lt M lt N Model 3b Allee effect Assumptions 1 If the population is too large then the growth is negative 2b If the population is too small then the growth rate per capita is positive but not the largest d PkPlt1 igtlt5 1gt dt N M k 2 growth rate per capita coefficient N carrying capacity M no meaning here and M lt O lt M lt N An alternative form dP kP 1 c P b 2 dt lt gt k maximum growth rate per capita b the population size when the growth rate per capita is the largest b gt 0 Case a 926 gt 1 Case b O lt b2clt 1 Example 1 A biologist start with 10 cells in a culture After each of the first four days he counts 25 62 154 and 375 cells Assuming a Malthus model what is the reproduction rate What will be the number of cells after 10 days Linear Regression The goal of linear regression is to adjust the values of slope and intercept to find the line that best predicts Y from X More precisely the goal of regression is to minimize the sum of the squares of the vertical distances of the points from the line Method of linear regression a get a set of data 331791 x2y2 mmyn b put it into a computer linear regression program Mechanism of linear regression 1 191 292 mmyn Goal Find a line y kxb minimize the sum of the squares of the vertical distances of the points from the line Vertical distance from xbyi to y 1m b kxi b yz Mathematical problem Find k and b which minimize fkb 2 km b M i1 n 8fn 8k i2kxz b 9031 07 8b i2kxz b 92 O o solve k and b from S n 33 k I anZyiand 11 2 M EM k I Z xiyi 11 1 Data fitting for Malthus model estimate is and P0 a Get a set of data t1P1 t2P2 tnPn b Take In to B5 let Q hP and get new data set t1Q1 t27Q21 1 tna C Put the data set t Qi to your linear regression program and get the output slope is and intercept b d In the solution of Malthus model Pt POekt we have In Pt In 130 kt so b In 130 Then PO eb So an exponential function which best fits the data is Pt ebkt where k and b are found in linear regression 8 Data fitting for Logistic model estimate k and N P0 is known P Notice that In 2 In 0 kt N P N a Get a set of data t1P1 t2P2 tnPn b Let Q InPZgt InN Pi and get new data set t1Q1 t27Q21 1 tn C Put the data set tiQi to your linear regression program and get the output slope k and intercept b d k is the maximum growth rate per capita in the Logistic P0 0 equation b In from which we can solve N provided PO is known

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