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1. Extinction of Neanderthals: Neanderthals were the original inhabitants of Europe and their species was very stable for more than 60,000 years. Forty thousand years ago the Neanderthals were replaced by our ancestors, the early humans (Cro-Magnon), who came to Europe from Africa much later. There is evidence that the two species coexisted in some parts of Europe. The mass extinction of the Neanderthals was rapid, in 5,000 to 10,000 years. Theories for the demise of the Neanderthals include genocide by the early humans and competition for resources with humans. We will investigate the latter possibility. Let N(t) be the total population of the humanoids, which consists of a population of Neanderthals, x(t), and early human, y(t): N(t) = x(t) + y(t). Suppose that they lived in the same resource-limited environment and therefore the total population satisfies the logistic equation: dN dt = rN(1 − N K ) − βN (1) where K is the total carrying capacity for all the humanoids combined, and β is their mortality rate. (We could have included β in the definition of r (and K), but we chose not to do so for convenience.) Assume r > β > 0 because the net growth rate should be positive for small population densities. (a) Suppose there is no difference in their survival skills. Write down two coupled equations for x(t), and y(t), in the form 1 x d dtx = F(x, y) − β (2) 1 y d dty = F(x, y) − β (3) where F(x, y) is the same for x and y. Find F(x, y). (Note that N = x + y satisfies the logistic equation (1). ) (b) Suppose the early humans are slightly better adapted to survival than Neanderthals, but the difference is tiny. Replace the human equation (3) with 1 y d dty = F(x, y) − (1 − )β (4) where 0 <   1 is the mortality difference. The Neanderthal equation remains the same. Find the equilibria of Eq. (2) and (4). (c) Determine the linear stability of the equilibria. (d) Discuss the implications of the results on equilibria and their stability. Is the extinction of the Neanderthals inevitable? (e) We know from paleontological data that it took 5,000 to 10,000 years for the Neanderthals to become extinct. Take 10,000 years as the time for x y to decrease by a factor of 100. Estimate the mortality difference , by forming an equation for d dt  x y  = 1 y dx dt −  x y  1 y d dty = · · · Suppose we measure β by the reciprocal of the lifetime of an individual, 30 years. 1 3. Stability of radiative equilibrium temperature: In the energy balance models for climate modeling, we will consider the following effects: • Incoming solar radiation - written as Qs(y) with y = sin θ, where θ is the latitude angle. In particular, we use s(y) = 1 − S2P2(y), where S2 = 0.482, P2(y) = 3y 2 − 1 2 • Albedo effect: A fraction of the solar radiation is reflected back to space without being absorbed by earth. The amount absorbed by earth per unit area is Q s(y)(1 − α(y)) where the value of albedo α(y) depends on the earth surface condition. According to ice dynamics, we’ll use α(y) = ( α2 = 0.62, y > ys α1 = 0.32 y < ys where ys is the location of the ice line. The poleward of this latitude the earth is covered with ice and equatorward of this location, the earth is ice-free. On the ice line, the temperature is taken to be Tc = −10◦C (ice sheet forms when T < Tc ): T(ys) = Tc, α(ys) = α0 = α1 + α2 2 = 0.47 • Outward radiation: The solar radiation warms up the earth, which in turn reemits energy back to space by mostly infrared radiation. It can be modeled as I = A + B T where A = 202 watts per square meter, and B = 1.90 watts per square meter per ◦C will be used. • Transport effect: there is energy exchange between different latitudes due to the temperature difference. The dynamical transport processes can be modeled by D(y) = C(T − T) where C = 1.6B will be assumed. (a) Let R be some constant such that R ∂ ∂tT is equal to the net solar energy input minus the outward radiation energy, plus the heat gained or lost from transport. Write down the model equation when the transport effect is ignored – the solution is called radiative solution. (b) Find the equilibrium solution, and the global mean temperature at equilibrium, i.e., find T ∗ and T ∗ . Note that, due to the symmetry assumption of northern and southern hemisphere, the global mean temperature can be computed as hemispherically averaged temperature: T = Z 1 0 T(y) dy You’ll need to first derive the model equation for T from the model equation in a). 3 (c) Discuss the linear stability of T ∗ for ice-free earth, snowball earth, and partially ice-covered earth. (d) Assuming they are all stable, find the globally averaged temperature at equilibrium for the above three scenarios. Use the present Q = 343 watts per square meter, and for the partially icecovered case, use the current ice line location at 72◦ latitude.

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