The possible control of competitive invasion by infection of the invader and multiplicative noise is studied. The basic model is the Lotka-Volterra competition system with emergent carrying capacities. Several stationary solutions of the non-infected and infected system are identified as well as parameter ranges of bistability. The latter are used for the numerical study of diffusive invasion phenomena. The Fickian diffusivities, the infection but in particular the white and colored multiplicative noise are the control parameters. It is shown that not only competition, possible infection and mobilities are important drivers of the invasive dynamics but also the noise and especially its color and the functional response of populations to the emergence of noise.
     The variability of the environment can additionally be modelled by applying Fokker-Planck instead of Fickian diffusion. An interesting feature of Fokker-Planck diffusion is that for spatially varying diffusion coefficients the stationary solution is not a homogeneous distribution. Instead, the densities accumulate in regions of low diffusivity and tend to lower levels for areas of high diffusivity. Thus, the stationary distribution of the Fokker-Planck diffusion can be interpreted as a reflection of different levels of habitat quality [1-5]. The latter recalls the seminal papers on environmental density, cf. [6-7]. Appropriate examples will be presented.

[1] Bengfort, M., Malchow, H., Hilker, F.M. (2016). The Fokker-Planck law of diffusion and pattern formation in heterogeneous media. Journal of Mathematical Biology 73(3), 683-704.
[2] Siekmann, I., Malchow, H. (2016). Fighting enemies and noise: Competition of residents and invaders in a stochastically fluctuating environment. Mathematical Modelling of Natural Phenomena 11(5), 120-140.
[3] Siekmann, I., Bengfort, M., Malchow, H. (2017). Coexistence of competitors mediated by nonlinear noise. European Physical Journal Special Topics 226(9), 2157-2170.
[4] Köhnke, M.C., Malchow, H. (2017). Impact of parameter variability and environmental noise on the Klausmeier model of vegetation pattern formation. Mathematics 5, 69 (19 pages).
[5] Bengfort, M., Siekmann, I., Malchow, H. (2018). Invasive competition with Fokker-Planck diffusion and noise. Ecological Complexity 34, 134-13.
[6] Morisita, M. (1971). Measuring of habitat value by the ``environmental density'' method. In: Spatial patterns and statistical distributions (Patil, C.D., Pielou, E.C., Waters, W.E., eds.), Statistical Ecology, vol. 1, pp. 379-401. Pennsylvania State University Press, University Park.
[7] N. Shigesada, N., Kawasaki, K., Teramoto, E. (1979). Spatial segregation of interacting species. Journal of Theoretical Biology 79, 83-99.