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We study the behaviour of the solutions of the Cauchy problem $$u_t={\left(u^m\right)}_{xx}+u\left(1-u^{m-1}\right),x\in \mathbb{R},t>0u\left(0,x\right)=u_0\left(x\right),u_0\left(x\right)\ge 0,$$ and prove that if initial data $u_0\left(x\right)$ decay fast enough at infinity then the solution of the Cauchy problem approaches the travelling wave solution spreading either to the right or to the left, or two travelling waves moving in opposite directions. Certain generalizations are also mentioned.