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In general, most differential equation models involve an idealized version of the real situation: nature itself is way too messy. Part of the art of mathematical modeling is to consider enough detail to capture the important effects, while ignoring irrelevant complications. A model of continuous matter ignores the fact that matter is made up of molecules ...


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Given a set of points in 3 dimensions there are multiple surfaces which might approximate the model. You could use one of these methods: Taylor series Divided differences Hermite interpolation Cubic spline interpolation Parametric Curves Bezier Curves I think that all of these have 3d versions although I have only used the 2d versions. What your maximum ...


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Since the system of two ODEs can be transformed to an Abel's differential equation of first kind, there is no known closed form for the solutions in the general case. However, closed forms might exist in some particular cases (i.e. for particular values of the parameters $a,b,c,d,k$ ).


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When the population is small, it will grow with a factor $e^r$ from one generation to the next. However, the environment (or other conditions) limits the population size. The population capacity of the system is $k$, so if the population grows beyond $k$ this cannot be maintained and the population will start decreasing. Also, as the population increases ...


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I have validated your solution with the following code: p1 = -20.2090; p2 = 17.3368; p3 = 272.9057; beta = 7.8*pi/180; x0 = 2000; c = @(z)4800+p1+p2*z(1)/1000+p3*exp(-.75*z(1)/1000); cp = @(z)(p2/1000-.75/1000*p3*exp(-.75*z(1)/1000)); q0 = (c(x0)/cos(beta))^2; ode = @(t,z)[z(2); -q0*cp(z)/c(z).^3]; IC = [x0; tan(beta)]; [x,z] = ode45(ode,[0 25*6076], IC); ...


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(a) Explain which is the predator species and which is the prey species. Does the model assume that the predator species has anything else to eat other than the prey? Explain. Well, in Volterra-type models predators and prey can be distinguished by how their population changes alone, without any other species. From point of view of dynamical systems ...


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In addition to Computational Fluid Dynamics, as you mention, which can also be used for oil and gas flows (and mud flow when drilling), there's also control theory and optimization, including flow networks to model pipe systems.



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