Mathematical Modelling with Case Studies: Using Maple and Subjects Mathematics & Statistics DownloadPDF MB Read online. , English, Book, Illustrated edition: Mathematical modelling with case studies : a differential equation approach using Maple / Belinda Barnes and Glenn R. Mathematical Modelling with Case Studies: Using Maple and MATLAB, Third Edition (3rd ed.) by CRC Press. Read online, or download in secure PDF format.

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Request PDF on ResearchGate | On Jan 1, , Richard Haberman and others published Mathematical Modelling with Case Studies: A Differential Equation. MATHEMATICAL MODELLING. WITH CASE STUDIES. A Differential Equations Approach. Using Maple™ and MATLAB®. Second Edition. Belinda Barnes. MATHEMATICAL MODELLING WITH CASE STUDIES. B. Barnes and International Standard Book Number (eBook - PDF). This book.

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Environmental management and manufacturing performance: the role of collaboration in the supply chain. Detecting art forgeries. This original rate of disintegration is too high. The painting is modern. The painting could be authentic. Cold pills. It does, however, increase the time taken for the drug to be removed from the blood stream. Accordingly, the peak concentration in the blood stream is higher. The maximum level in the bloodstream is approximately 6.

The amount of tetracycline in the GI-tract levels out at just over 1 milligram after approximately 5 hours. The urinary tract is considered to be an absorbing compartment and so it is not surprising that the amount of tetracycline continues to rise. Alcohol consumption. If the man consumes two drinks his BAL just exceeds the legal limit of 0.

The BAL of both women and men is lower if alcohol is consumed after a substantial meal than on an empty stomach, as expected. After a meal, women and men can consume three standard drinks at once and remain under the limit.

The response to the initial data is x0 t0 t. There is no response to the input. Formulating DEs for alcohol case study. Economic model. There- after, the capital—labour ratio decreases with labour becoming redundant, and this redundancy growing.

Return to scale property. Stability of equilibrium solution. Thus r 1 e is a stable steady state. By similar arguments, we see that r 2 e is unstable and r 3 e is stable. Stability of the steady states of r t for Question Modelling the spread of technology. Density dependent births. Mouse population model. Harvesting model. Fishing with quotas. Predicting with a model. The population reaches approximately after 2 months.

Plant biomass. Modelling the population of a country. Let the immigration rate be i and the emigration rate be e. Let x t be the population in country X at time t and y t be the population in the neighbouring country Y at time t. Subscripts x and y denote the rates in countries X and Y , respectively. Sensitivity to initial conditions. Investigating parameter change. Similar behaviour to the logistic model is exhibited.

With higher r values, the steady state is again reached, although damped oscillations are exhibited. For r near 3 solutions exhibit damped oscillations towards the non-zero steady state.

Aquatic environments. Therefore, the function is bounded from above. Stability of 2-cycles. Numerical schemes. All numerical solutions use a stepsize of 0. Numerical and analytical solutions. See Figures 5 and 6. All numerical solutions use use a stepsize of 0. All nu- merical solutions use a stepsize of 0. When Digits is set to 20, the error decreases as the stepsize increases. With Digits set to 10, however, it appears that the error is larger when a stepsize of 0.

Numerical solution of basic epidemic model. The number of susceptibles who never become infected falls to approximately 2. Contagious for life.

Disease with no immunity. The word-equations are rate of change in no. Continuous vaccination. SEIR model, disease with latent period. Two prey and one predator. Furthermore, the equilibrium predator population decreases but the equilibrium prey population increases.

Predator-prey with density dependence. Competing species with no density dependence. Competing species with density dependence.

Simple age-based model. Beetle population model. Compartment diagram for Question 13 with larvae, pupae and adult beetles. Wine fermentation.

Let the density of yeast cells, the amount of alcohol and the amount of sugar at time t be Y t , A t and S t , respectively. Cycles of measles epidemics. Density dependent contact rate. Battle loss due to disease. Jungle warfare. Exact solution for battle model. Spread of malaria by mosquitoes. There are four compartments, susceptible mosquitoes and humans and in- fected mosquitoes and humans. Note that there is no recovery for either humans or mosquitoes. In this model we have also ignored any births and deaths of mosquitoes or humans.

Infections can only occur by a mosquito biting a human. Compartment diagram for Question Spread of a religion. In the simplest model, unconverted become converted and converted become missionaries. A compartment diagram is shown below. Predator-prey with protection of young prey. The natural death rate of adult prey is d and the rate of death by predation is eY. The per-capita birth and death rates of predators are gX2 and f, respectively. Diseases with carriers. There are others.

Basic reproduction number. The value of R0 is smaller than for the SIR model because this model accounts for some people dying of natural causes before they recover for the disease. Farmers, bandits and soldiers.

However, it will take longer for it to happen. Simple example. Finding equilibrium points. Using the chain rule. A phase-plane trajectory is shown in Figure 9. S I Figure 9: A typical trajectory direction for Question 4. Disease with reinfection. These directions are shown in Figure A typical trajectory direction for Question 5. Predator-prey with density dependent growth of prey.

Predator-prey with DDT. This is in accordance with the results of Figure 5. Predator-prey with density dependence and DDT. One prey and two predators. Competing species without density dependence. The phase-plane diagram is shown in Figure 11 27 Regions in the phase-plane illustrating the trajectory directions for Question The dashed lines are the non-zero nullclines.

Rabbits and foxes. A typical phase-plane trajectory is shown in Figure A typical trajectory for Question The dashed line is the non-zero nullcline. From Figure 12 it is clear that intersection with the Y axis is possible. This corresponds to the rabbits completely dying out. Microorganisms and toxins. The amount of toxin always increases. The number of microorganisms initially increases but then decreases as more toxin is produced. Fatal disease. This is what we would expect intuitively.

This higher transmission rate causes more susceptibles to become infected. Battle model with desertion. Since both R and B can only decrease, possible trajectories are shown in Figure B R Figure Battle with long range weapons.

At the end of the epidemic, suppose there are sf susceptibles left and no infectives. Bacteria in gut. Constructing a phase-plane. Phase-plane for the system in Question 1. Phase-plane for the system in Question 3. See Figure 19 for a typical phase-plane diagram. Phase-plane for the system in Question 4. The determinant of J, evaluated at the equilibrium point, is the product of the eigenvalues and so we must show that the determinant is negative.

Regardless of the value of q the determinant will be negative at one of the equilibrium points and so one of the equilibrium points will be a saddle point. Linearisation of competition model. The species will coexist. Note that the steady state representing coexistence is not feasible. Comparison of nonlinear and linearised models.

Time-dependent plots for the nonlinear and linearised systems are shown in Figure For initial values close to the non-zero equilibrium solution of , , the solution obtained using the linearised system is similar to the full solution.

Prey a and predator b populations over time computed using the linearised system grey line and the full system black line for Ques- tion 6. The initial prey and predator populations are , and 90,

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