Allometric growth, disturbance regime, and dilemmas of controlling invasive plants: a model analysis

Abstract

Disturbed communities are observed to be more susceptible to invasion by exotic species, suggesting that some attributes of the invaders may interact with disturbance regime to facilitate invasion success. Alternanthera philoxeroides, endemic to South America, is an amphibious clonal weed invading worldwide. It tends to colonize disturbed habitats such as riparian zones, floodplain wetlands and agricultural areas. We developed an analytical model to explore the interactive effects of two types of physical disturbances, shoot mowing and root fragmentation, on biomass production dynamics of A. philoxeroides. The model is based on two major biological assumptions: (1) allometric growth of root (belowground) vs. shoot (aboveground) biomass and (2) exponential regrowth of shoot biomass after mowing. The model analysis revealed that the interaction among allometric growth pattern, shoot mowing frequency and root fragmentation intensity might lead to diverse plant ‘fates’. For A. philoxeroides whose root allocation decreases with growing plant size, control by shoot mowing was faced with two dilemmas. (1) Shoot regrowth can be effectively suppressed by frequent mowing. However, frequent shoot mowing led to higher biomass allocation to thick storage roots, which enhanced the potential for faster future plant growth. (2) In the context of periodic shoot mowing, individual shoot biomass converged to a stable equilibrium value which was independent of the root fragmentation intensity. However, root fragmentation resulted in higher equilibrium population shoot biomass and higher frequency of shoot mowing required for effective control. In conclusion, the interaction between allometric growth and physical disturbances may partially account for the successful invasion of A. philoxeroides; improper mechanical control practices could function as disturbances and result in exacerbated invasion.

Appendix 1: Derivation of the general equations and the condition for S n = S n−1 and P n = P n−1

At individual level

Individual shoot biomass at the first mowing (time 0, see Fig. 2) is S ₀. According to Eq. 5, shoot biomass at the second mowing (time T), S ₁, is

$$ S_1 = kaS_0^b {\text{e}}^{rT} $$

Similarly, at the third mowing (time 2T), S ₂ is

$$ S_2 = kaS_1^b {\text{e}}^{rT} = k^{b + 1} a^{b + 1} S_0^{b^2 } {\text{e}}^{(b + 1)rT} $$

and at the fourth mowing (time 3T), S ₃ is

$$ S_3 = kaS_2^b {\text{e}}^{rT} = k^{b^2 + b + 1} a^{b^2 + b + 1} S_0^{b^3 } {\text{e}}^{(b^2 + b + 1)rT} $$

Conclusively, individual shoot biomass at the (n + 1) th mowing (time nT), S _n, is given by

$$ \left\{ {\begin{array}{ll} {S_{\text{n}} = (ka{\text{e}}^{rT} )^{\frac{{1 - b^{\text{n}} }}{{1 - b}}} S_0^{b^{\text{n}} } } & {{\text{if }}b \ne 1} \\ {S_{\text{n}} = (ka{\text{e}}^{rT} )^{\text{n}} S_0 } & {{\text{if }}b = 1} \\ \end{array} } \right. $$

Now the condition under which S _n = S _n−1 turns out to be equal to

$$ \left\{ {\begin{array}{*{20}c} {(ka{\text{e}}^{rT} ) ^{\frac{{1 - b^{\text{n}} }}{{1 - b}}} S_0^{b^{\text{n}} } \, = (ka{\text{e}}^{rT} )^{\frac{{1 - b^{{\text{n - 1}}} }}{{1 - b}}} S_0^{b^{{\text{n - 1}}} } } & {{\text{if }}b \ne 1} \\ {(ka{\text{e}}^{rT} )^{\text{n}} S_0 = (ka{\text{e}}^{rT} ) ^{{\text{n - 1}}} S_0 } & {{\text{if }}b = 1} \\ \end{array} } \right. $$

This expression can be simplified to

$$ ka{\text{e}}^{rT} S_0^{b - 1} = 1 $$

After incorporating Eq. 1 into the above equation, we can obtain Eq. 9:

$$ ka^{\frac{1}{b}} {\text{e}}^{rT} R_0^{\frac{{b - 1}}{b}} = 1 $$

At population level

Population shoot biomass at the first mowing (time 0) is P ₀. According to Eq. 7, population shoot biomass at the second mowing (time T), P ₁, is

$$ P_1 = Nka\left(\frac{{P_0 }}{N}\right)^b {\text{e}}^{rT} = N^{1 - b} kaP_0^b {\text{e}}^{rT} $$

Similarly, at the third mowing (time 2T), P ₂ is

$$ P_2 = Nka\left(\frac{{P_1 }}{N}\right)^b {\text{e}}^{rT} = N^{1 - b^2 } k^{b + 1} a^{b + 1} P_0^{b^2 } {\text{e}}^{(b + 1)rT} $$

and at the fourth mowing (time 3T), P ₃ is

$$ P_3 = Nka(\frac{{P_2 }}{N})^b {\text{e}}^{rT} = N^{1 - b^3 } k^{b^2 + b + 1} a^{b^2 + b + 1} P_0^{b^3 } {\text{e}}^{(b^2 + b + 1)rT} $$

In conclusion, the population shoot biomass at the (n + 1) th mowing (time nT), P _n, is given by

$$ \left\{ {\begin{array}{ll} {P_{\text{n}} = N^{1 - b^{\text{n}} } (ka{\text{e}}^{rT} )^{\frac{{1 - b^{\text{n}} }}{{1 - b}}} P_0^{b^{\text{n}} } } & {{\text{if }}b \ne 1} \\ {P_{\text{n}} = (ka{\text{e}}^{rT} )^{\text{n}} P_0 } & {{\text{if }}b = 1} \\ \end{array} } \right. $$

Now the condition under which P _n = P _n−1 is equal to

$$ \left\{ {\begin{array}{*{20}c} {N^{1 - b^{\text{n}} } (ka{\text{e}}^{rT} )^{\frac{{1 - b^{\text{n}} }} {{1 - b}}} P_0^{b^{\text{n}} } \quad= N^{1 - b^{{\text{n - 1}}} } (ka{\text{e}}^{rT} )^{\frac{{1 - b^{{\text{n - 1}}} }}{{1 - b}}} P_0^{b^{{\text{n - 1}}} } } & {{\text{if }}b \ne 1} \\ {(ka{\text{e}}^{rT} )^{\text{n}} P_0 = (ka{\text{e}}^{rT} ) ^{{\text{n - 1}}} P_0 } & {{\text{if }}b = 1} \\ \end{array} } \right. $$

This expression can be simplified to

$$ ka{\text{e}}^{rT} (\frac{{P_0 }}{N})^{b - 1} = 1 $$

further to

$$ ka{\text{e}}^{rT} S_0^{b - 1} = 1 $$

After incorporating Eq. 1 into the above equation, we arrive at Eq. 9 again:

$$ ka^{\frac{1}{b}} {\text{e}}^{rT} R_0^{\frac{{b - 1}}{b}} = 1 $$