# American Institute of Mathematical Sciences

November  2020, 25(11): 4411-4426. doi: 10.3934/dcdsb.2020104

## Modelling fungal competition for space:Towards prediction of community dynamics

 1 ORCID: 0000-0003-4452-7106, Department of Mathematics, Swansea University, Bay Campus, Swansea, SA1 8EN, UK 2 ORCID: 0000-0003-1544-0407, Department of Biosciences, Swansea University, Singleton Park Campus, Swansea, SA2 8PP, UK 3 ORCID: 0000-0003-0486-5450, Department of Mathematics, Swansea University, Bay Campus, Swansea, SA1 8EN, UK

* Corresponding author: Chenggui Yuan

Received  October 2019 Revised  November 2019 Published  April 2020

Fund Project: DAK was supported by a student scholarship from Swansea University

Filamentous fungi contribute to ecosystem and human-induced processes such as primary production, bioremediation, biogeochemical cycling and biocontrol. Predicting the dynamics of fungal communities can hence improve our forecasts of ecological processes which depend on fungal community structure. In this work, we aimed to develop simple theoretical models of fungal interactions with ordinary and partial differential equations, and to validate model predictions against community dynamics of a three species empirical system. We found that space is an important factor for the prediction of community dynamics, since the performance was poor for models of ordinary differential equations assuming well-mixed nutrient substrate. The models of partial differential equations could satisfactorily predict the dynamics of a single species, but exhibited limitations which prevented the prediction of empirical community dynamics. One such limitation is the arbitrary choice of a threshold local density above which a fungal mycelium is considered present in the model. In conclusion, spatially explicit simulation models, able to incorporate different factors influencing interaction outcomes and hence dynamics, appear as a more promising direction towards prediction of fungal community dynamics.

Citation: Diogenis A. Kiziridis, Mike S. Fowler, Chenggui Yuan. Modelling fungal competition for space:Towards prediction of community dynamics. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4411-4426. doi: 10.3934/dcdsb.2020104
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Examples from the empirical system of fungal mycelia. (a) One-species setting of an extending mycelium inoculated at the centre of the 22.4 $\times$ 22.4 cm dish. (b) Transparency of drawn boundaries of three-species mycelia inoculated randomly at 49 sites indicated by the letters H (for Hf), T (Tv) and V (Vc). (c) The transparency of the mycelial boundaries of (b), processed in ImageJ for obtaining species relative occupancy at time $t = 0$ d
One-species empirical dynamics compared to the model dynamics from the ODE (17). Circles are for the Tv empirical relative cover of the dish, and curves are for the relative abundance of Tv in a well-mixed culture with the ODE model. (a) A single Tv mycelium inoculated at the centre of a dish in the empirical system, extending before reaching the edges of the dish. (b) Three Tv mycelia closely inoculated at the dish centre, fusing to form one mycelium which extended and covered the dish. The initial conditions were equal to the relative cover of Tv on the empirical dishes at time $t = 0$ d. The extension rate parameter $e_A \approx 0.35$ d$^{-1}$
). (b) Three Tv mycelia closely inoculated at the dish centre (inset is a numerical solution of the PDE model at $t = 12$ d), fusing to form one mycelium which extended and covered the dish (as in Fig. 2b). The PDE model for Tv had initial conditions similar to the empirical setting, with growth rate $\epsilon_A = 2.16$ d$^{-1}$, and diffusion coefficient $\delta_A = 0.017$ cm$^2$ d$^{-1}$. It was assumed that a mycelium is present when its density $A > 0.01$. The relative cover in the PDE solution was estimated by Monte Carlo integration of the mycelium present (the curve in each panel is 95% confidence region of the mean relative cover in the PDE solution from 100 Monte Carlo integrations)">Figure 3.  One-species empirical dynamics compared to the model dynamics with the one-species PDE (34). Circles are for the Tv empirical relative cover of the dish, and curves are for Tv from the PDE model. (a) A single Tv mycelium inoculated at the centre of a dish in the empirical system (inset is the PDE model's solution at $t = 12$ d), extending before reaching the edges of the dish (as in Fig. 2a). (b) Three Tv mycelia closely inoculated at the dish centre (inset is a numerical solution of the PDE model at $t = 12$ d), fusing to form one mycelium which extended and covered the dish (as in Fig. 2b). The PDE model for Tv had initial conditions similar to the empirical setting, with growth rate $\epsilon_A = 2.16$ d$^{-1}$, and diffusion coefficient $\delta_A = 0.017$ cm$^2$ d$^{-1}$. It was assumed that a mycelium is present when its density $A > 0.01$. The relative cover in the PDE solution was estimated by Monte Carlo integration of the mycelium present (the curve in each panel is 95% confidence region of the mean relative cover in the PDE solution from 100 Monte Carlo integrations)
Prediction of the three-species empirical community dynamics with the PDE (35–37). The data points are for the empirical relative cover of the species in time, and the curves are for the PDE model. The inset shows the PDE numerical solution at time $t = 40$ d. Colour–point (of each species): red–circle (Hf), cyan–square (Tv), and blue–$\times$ (Vc). The PDE model had initial conditions similar to the empirical setting, with the following parameter values (species A was Hf, B was Tv, and C was Vc): $\epsilon_A = 0.78$ d$^{-1}$, $\epsilon_B = 2.16$ d$^{-1}$, $\epsilon_C = 1.14$ d$^{-1}$, $\delta_A = 0.0062$ cm$^2$ d$^{-1}$, $\delta_B = 0.017$ cm$^2$ d$^{-1}$, $\delta_C = 0.0091$ cm$^2$ d$^{-1}$, $\rho_A = 0.22$ d$^{-1}$, and $\rho_B = 0$ d$^{-1}$. Local extension and replacement rates were taken from the mean boundary extension and replacement rates in the empirical dishes, as for the extension in the one-species PDE. It was assumed that a mycelium is present when its density is greater than 0.5. The relative cover in the PDE solution was estimated by Monte Carlo integration of the mycelium present (the curves are 95% confidence regions of the mean relative cover in the PDE solution from 100 Monte Carlo integrations)
Summary of the features and performance of the models considered in the present work
 Model Advantages Disadvantages Predictability ODE 1. Simplicity 1. Non-spatial 1. Low for even one-species 2. Determinism 3. Math. liability PDE 1. Spatial 1. Arbitrary mycelial presence (density threshold)2. Challenging set-up of exact initial conditions 3. Challenging measurement-parameterisation 4. Cannot model each mycelium separately 1. High for one-species2. Low for three-species 2. Determinism 3. Math. liability
 Model Advantages Disadvantages Predictability ODE 1. Simplicity 1. Non-spatial 1. Low for even one-species 2. Determinism 3. Math. liability PDE 1. Spatial 1. Arbitrary mycelial presence (density threshold)2. Challenging set-up of exact initial conditions 3. Challenging measurement-parameterisation 4. Cannot model each mycelium separately 1. High for one-species2. Low for three-species 2. Determinism 3. Math. liability
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