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I have been trying to approach a modelling case about organism propagation which reproducing with velocity $$\alpha$$ spreading randomly according these equations:

$$\frac{du(x,t)}{dt}=k\frac{d^2u}{dx^2} +\alpha u(x,t)\\\ \\ u(x,0)=\delta(x)\\\ \lim\limits_{x \to \pm\infty} u(x,t)=0$$

This studying case requires to demonstrate that isoprobability contours, it means, in the points (x,t) which P(x,t)=P=constant is verified that

$$\frac{x}{t}=\pm [4\alpha k-2k\frac{\log(t)}{t}-\frac{4k}{t}\log(\sqrt{4\pi k} P)]^\frac{1}{2}$$

Another aspect to demonstrate is that $t \to \infty$, the spreading velocity of these contours, it means, the velocity which these organisms are spreading is aproximated to

$$\frac{x}{t}\pm(4\alpha k)^\frac{1}{2}$$

Finally, how to compare this spreading velocity with purely diffusive process $(\alpha=0)$, it means , x is aproximated to $$\sqrt{kt}$$

This is just for academical curiosity and I would like to understand better this kind of cases with Partial Differential Equations. So, I require any guidance or starting steps or explanations to find the solutions because it's an interesting problem.

Thanks very much for your attention.