# Preface

This tutorial was made solely for the purpose of education and it was designed
for students taking Applied Math 0340. It is primarily for students who
have some experience using *Mathematica*. If you have never used
*Mathematica* before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA
0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The *Mathematica* commands in this tutorial are all written in **bold black font**, while *Mathematica* output is in regular fonts.

Finally, you can copy and paste all commands into your *Mathematica* notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License
(GPL). You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately. The tutorial accompanies the
textbook *Applied Differential Equations.
The Primary Course* by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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Introduction to Linear Algebra with *Mathematica*

## Glossary

# Miscellany

**Example 1: **
The scaled equations (see Hal L. Smith and Paul Waltman, *The Theory of the Chemostat: Dynamics of Microbial Competition*, Cambridge University Press, 1995) for the growth of two populations of microorganisms *x*_{1} and *x*_{2}, with corresponding parameters *a*_{i} and *m*_{i}, *i*=1,2, in a chemostat are

*S*(

*t*) denotes the concentration of the nutrient at time

*t*. Letting Σ = a -

*S*-

*x*

_{1}-

*x*

_{2}, we can see that

*break-even*concentrations

_{1}is different from λ

_{2}, then there is no "interior" rest point, that is, a rest point with both components positive. The only interesting cases are where

*m*

_{i}> 1 and 0 < λ

_{i}< 1 for

*i*= 1,2.

**Theorem:**Suppose that

*m*

_{i}> 1,

*i*= 1,2, and that 0 < λ

_{1}< λ

_{2}< 1. Then any solution of the above chemostat system with

*x*

_{i}> 0 satisfies

**Example 2: **
A standard single-species logistic model with continuous constant dispersal rate between two heterogeneous patches can be written as follows

*N*

_{1},

*N*

_{2}represents the population density in the corresponding habitat at time

*t*,

*r*

_{i}and

*M*

_{i}are the intrinsic rate of population increase and the carrying capacity of population with index

*i*, δ is the fraction of migrants dying during migration and

*m*is the emigration rate, a constant. Above dispersal model may be used to characterize the mobility of bird or, insect ■

**Example 3: **
N. Gregory Mankiw, David Romer, and David Weil created a human capital augmented version of the Solow–Swan model that can explain the failure of international investment to flow to poor countries. In this model output and the marginal product of capital (K) are lower in poor countries because they have less human capital than rich countries.

The mode is augmentation by human capital shares some similarities with the basic model but, on the other hand, it solves some problems imposed by the latter. They are summarized as follows: There is still no long run growth of GDP per capita due to the presence of decreasing returns to capital accumulation. There is still no role for technological progress or the latter, if any, cannot be explained. Convergence to the steady-state is slower as human capital accumulation partly offsets the impact of decreasing returns to physical investments. It is equivalent to allowing for a broader capital share in the Solow model. The slower convergence rate is in conformity with the real-life pattern.

On the empirical side, the MRW model is usually solved by a restricted
regression equation that relates the GDP per capita or working age population
to the inputs of production. The saving rate *s* is measured by the share of overall
or business *k* investments in GDP. The parameter *s* is calculated h as the average
percentage of the working-age population in secondary school over a long time
period. Mankiw et al. relate the latter over the period 1960-1985 to GDP per
working age person in 1985.

Similar to the textbook Solow–Swan model, the production function is of Cobb–Douglas type:

*n*and

*g*.

Thus *k*, denoting the stock of human capital and *h*, representing
the stock of human capital can be derived, and then modified to
represent the steady state.

- Mankiw, N.G., Romer, D., Weil, D.N., A contribution to the empirics of economic growth, The Quaterly Journal of Economics, 1992, May, pp. 407--437.

**Example 4: **
The **Ramsey–Cass–Koopmans model**, or Ramsey growth model, is a neoclassical model of economic growth based primarily on the work of Frank P. Ramsey (1903--1930), with significant extensions by David Cass (1937--2008) and Tjalling Koopmans (1910--1985).

Let *f*(*k*) = *k*^{α} be
a production function measuring therelative economic output in terms of *k* and a capital elasticity parameter α (the responsiveness of the output production to changes in the input capital).

ϕ be the growth rate of labor productivity (for example, due to technological innovation or efficiency improvements).

ξ be the growth rate of labor supply (for example, due to migration or population increase).

δ be the depreciation rate of capital (for example,due to inflation).

θ be an elasticity parameter indicating the tendencyof consumers to smooth out their consumption over time.

ρ be the rate at which consumers discount theirfuture consumption (for example, by indicating a preference for immediate consumption or attempting to preserve their long-term average onsumption).

The Ramsey-Cass Koopmans Model is composed of two coupled differential equations

*k*, with respect to time. The latter models the rate of change of per capita consumption,

*c*, with respect to time.

We can find its steady state (*k**, *c** ) by setting both derivatives to zero. This when solved would lead us to the steady state equations of \( f_k (k^{\ast}) = \delta + \rho \) and \( c^{\ast} = f(k^{\ast}) - \left( \phi + \delta \right) k^{\ast} . \)

The Jacobian of this system of equations becomes

- Cass, David (1965). "Optimum Growth in an Aggregative Model of Capital Accumulation". Review of Economic Studies. 32 (3): 233–240. doi:10.2307/2295827
- Gourinchas, P.-O., The Ramsey-Cass-Koopmans Model, Berkely.
- Koopmans, T. C. (1965). "On the Concept of Optimal Economic Growth". The Economic Approach to Development Planning. Chicago: Rand McNally. pp. 225–287.
- Ramsey, Frank P. (1928). "A Mathematical Theory of Saving". Economic Journal. 38 (152): 543–559. doi:10.2307/2224098

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