Even if I have never seen this particular PDE, it has some terms resembling the viscous Burgers equation.

Do you get a numerical solution ? what is the problem with the plot?

Many thanks, Dr. Denaro, I did get the numerical solution, I donot know how to solve it numerically, and plot the graph.

The solution can be plotted in terms of the 1d function u(x) for several time steps and I suggest to use matlab. You can also plot the surface u(x,t) If you store the solutions in a matrix. 

Dear Dr. Denaro, thanks for your kindful help, however, I have no idea how to numerically solve and  plot it in matlab, could you give me some introduction?

http://mathcenter.hust.edu.cn/Uploads/file/20160128/56a98740d59f8.pdf

Thanks，Dr. Denaro, I will read the useful material !

For me this equation looks a bit strange. On one hand, if you forget terms without derivative in the right part and recall values of parameters, it is almost heat equation: u_t+u_x=u_xx. Note that linear change of variables turn it to exact heat equation.

If the term u*u_x would be not small, then it indeed reminds me of Burgers equation. It can also describe nonlinear surface wave in water with dissipation. Given that you suggest some parameters, you might be a physicist. But what is the physical sense of terms at right part? Why no small parameter is assigned to them?

As for existence of periodic solution, you can try it analytically. Just look for solutions in a form u(x,t)=u(x-vt), take derivatives, and get not PDE, but ordinary DE, that often can be integrated analytically.

I agree that this pde looks strange. If it were dimensional it would turn simply wrong....

I would have mores details about the physical problem it describes...

Thanks !, Dr. Yuri Yegorov 

Xianbo, if you are interested in dynamical systems in biology, the main idea (apart from logistic equation) is Lotka-Volterra system. See https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations . Here you can find about combination of Lotka-Volterra and logistic eq.: https://en.wikipedia.org/wiki/Competitive_Lotka%E2%80%93Volterra_equations .There is also literature on shallow lake dynamics, for example, https://www.researchgate.net/profile/Florian_Wagener/publication/4831533_Skiba_points_and_heteroclinic_bifurcations_with_application_to_the_shallow_lake_system/links/02e7e5270e1dd62194000000/Skiba-points-and-heteroclinic-bifurcations-with-application-to-the-shallow-lake-system.pdf

But I do not know PDE for ecosystem. Perhaps it exists but you should search for it. Or you can derive yourself based on some assumptions.

I can recommend you papers of Vakulenko. For example, https://www.researchgate.net/publication/282603522_New_perspectives_stability_and_complex_dynamics_of_food_webs_via_Hamiltonian_methods

Article New perspectives: stability and complex dynamics of food web...

Article Skiba points and heteroclinic bifurcations, with application...

Dera Prof. Yegorov ,

many thanks for your sharing paper, they are indded very useful for me, especially, the first one. thanks ! looking forward to discuss with you further.

Using Malfliet's 'tanh' method at least 15 solutions, of which 11 are non-trivial, can be found.

Refs:

Malfliet, W., & Hereman, W. (1996). The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica Scripta, 54, 563–568. Retrieved from http://www.iop.org/EJ/abstract/1402-4896/54/6/003

Malfliet, W., & Hereman, W. (1996). The tanh method: II. Perturbation technique for conservative systems. Physica Scripta, 54, 569–575. Retrieved from http://www.iop.org/EJ/abstract/1402-4896/54/6/004

Graham W Griffiths and William E Schiesser (2011), Traveling Wave Analysis of Partial Differential Equations  - Numerical and Analytical Methods with Matlab and Maple, Academic Press (ISBN: 9780123846525)

Other methods, such as the 'exp' and 'Riccati' methods, also produce TW solutions.

Download some software that can solve the problem from link below.

http://www.pdecomp.net/