Spatiotemporal diffusion as early warning signal for critical transitions
in spatial tumor-immune system with stochasticity

Zhiqin Ma,1 Yuhui Luo,1 Chunhua Zeng,1, 2, ∗ and Bo Zheng3, 2
1Faculty of Science, Kunming University of Science and Technology, Kunming 650500, China
2Department of Physics, Zhejiang University, Hangzhou 310027, Peoples Republic of China
3School of Physics and Astronomy, Yunnan University, Kunming 650091, Peoples Republic of China

Complex dynamical systems have tipping points and exhibit nonlinear dynamics. It is difficult to predict and prevent the onset and progression of the tumors, mainly due to the complexity of interactions between tumor growth and tumor-immune cells involved. Moreover, previous models were based on the influence of the zerodimensional systems and did not consider the spatiotemporal fluctuation in the tumor microenvironment. In this paper, we extend the previous model to a two-dimensional system and use spatial early warning signals to study the spatially extended tumor-immune system with stochasticity. On the one hand, we obtain the stationary probability density of the system under the mean-field approximation assumption. It is found that the health
state gets more and more stable than the disease state as the noise level increases when the system has a bistable state, and the system undergoes from health to disease state through a bistable region as the growth rate increases. On the other hand, we present a spatiotemporal diffusion coefficient indicator to predict upcoming critical transitions. It is shown that a rising spatiotemporal diffusion coefficient obtained from the spatial snapshot data can be an effective indicator for predicting upcoming critical transitions. Anticipating critical transitions in the spatial tumor-immune system with stochasticity can be greatly helpful to prevent disease onset and progression, which may intercept abrupt shifts from health to disease state.