Bayesian Population Dynamics of Spreading Species

Abstract

The invasion by spreading species is one of the most serious threats to biodiversity and ecosystem functioning. Despite a number of empirical and theoretical studies, there is still no general model about why or when settlement becomes an invasion. The purpose of this work is to test a model of Bayesian population dynamics relying on best-response strategies that could help in resource management and bioeconomic modeling. Given the species survival probability, our static game unveils a breaking-level probability in mixed strategies, where the best response for exotic species is to invade and the best response for native species is to resist. In a dynamic setting, we introduce a stochastic version of the balance equation based on conditional probabilities. We find that when the species survival probability and the availability of resources in the ecosystem are respectively high and low, the population rebalancing dynamics operates at a high pace.

Appendix

$$ \begin{array}{c}\hfill {s}_{t+1}=1-{s}_t-{\overline{s}}_t \Pr \left[{\rho_{\overline{s}}}_s\Big|{s}_t\right]+{s}_t \Pr \left[{\rho}_{s\overline{s}}\Big|{\overline{s}}_t\right]\hfill \\ {}\hfill =1-{s}_t-{\overline{s}}_t \Pr \left[{s}_t\left(1-\tilde{q}\right)\Big|{s}_t\right]+{s}_t \Pr \left[{\overline{s}}_t\tilde{q}\Big|{\overline{s}}_t\right]\hfill \\ {}\hfill =1-{s}_t-{\overline{s}}_t\;{\scriptscriptstyle \frac{{\displaystyle {\int}_{1-\tilde{q}}^1{s}_t d{s}_t}}{{\displaystyle {\int}_0^1{s}_t d{s}_t}}}+{s}_t{\scriptscriptstyle \frac{{\displaystyle {\int}_{\tilde{q}}^1{\overline{s}}_t d{\overline{s}}_t}}{{\displaystyle {\int}_0^1{\overline{s}}_t d{\overline{s}}_t}}}\hfill \\ {}\hfill ={s}_t\left(2\tilde{q}-2{\tilde{q}}^2\right)+{\left(\tilde{q}-1\right)}^2\hfill \end{array} $$

(18)

$$ \begin{array}{c}{r}_{t+1}={r}_t+{\overline{r}}_t \Pr \left[{\rho}_{\overline{r} r}\Big|{r}_t\right]-{r}_t \Pr \left[{\rho}_{r\overline{r}}\Big|{\overline{r}}_t\right]\\ {}={r}_t+{\overline{r}}_t \Pr \left[{r}_t\left(1-\tilde{p}\right)\Big|{r}_t\right]-{r}_t \Pr \left[{\overline{r}}_t\tilde{p}\Big|{\overline{r}}_t\right]\\ {}={r}_t+{\overline{r}}_t\;{\scriptscriptstyle \frac{{\displaystyle {\int}_{1-\tilde{p}}^1{r}_t d{r}_t}}{{\displaystyle {\int}_0^1{r}_t d{r}_t}}}-{r}_t{\scriptscriptstyle \frac{{\displaystyle {\int}_{\tilde{p}}^1\overline{r} d{\overline{r}}_t}}{{\displaystyle {\int}_0^1{\overline{r}}_t d{\overline{r}}_t}}}\\ {}={r}_t\left(2{\tilde{p}}^2-2\tilde{p}\right)+2\tilde{p}-{\tilde{p}}^2\end{array} $$

(19)

1.1 System 17

Solving the dynamical equation ${s}_{t+1}={s}_t\left(2\tilde{q}-2{\tilde{q}}^2\right)+{\left(\tilde{q}-1\right)}^2$ reduces to solving the nonhomogeneous recurrence relation ${s}_t={c}_1\left(2\tilde{q}-2{\tilde{q}}^2\right)+{\left(\tilde{q}-1\right)}^2$.Withinthisrelation, ${s}_t={c}_1\left(2\tilde{q}-2{\tilde{q}}^2\right)$ is the associated homogeneous recurrence relation, which solution is ${s}_t={c}_1{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}$. The nonhomogeneous part c ₂ yields ${c}_2={\scriptscriptstyle \frac{{\left(\tilde{q}-1\right)}^2}{1-{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}}}$ from which we obtain ${c}_1={\scriptscriptstyle \frac{{\left(\tilde{q}-1\right)}^2}{\left[1-{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}\left]\kern0.5em \right[{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^t-{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}\right]}}$ and finally ${s}_t^{*}={\scriptscriptstyle \frac{{\left(\tilde{q}-1\right)}^2{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^t}{\left[1-{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}\left]\kern0.5em \right[{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^t-{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^{t-1}\right]}}$. The final condition is ${s}_t^{\prime }>0\iff {\scriptscriptstyle \frac{-4{\left(\tilde{q}-1\right)}^4{\tilde{q}}^2{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^t \ln \left(2\tilde{q}-2{\tilde{q}}^2\right)}{\left(2{\tilde{q}}^2-2\tilde{q}+1\right){\left[{\left(2\tilde{q}-2{\tilde{q}}^2\right)}^t+2{\tilde{q}}^2-2\tilde{q}\right]}^2}}>0$. The same rationale applies to r _t + 1.

1.2 Proposition 3 and Proposition 4

We construct the Jacobian matrix for the dynamical system D t + 1 from the partial derivatives over the availability of resources and the survival rate. The eigenvalues of the Jacobian matrix of five steady-state configurations S yield as follows.

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{l}2\left( s-1\right) a-\left(2 s-1\right) b-\lambda \kern2em 2\left( s-1\right) f-\left(2 s-1\right) d\hfill \\ {}-2\left( r-1\right) a+\left(2 r-1\right) b\kern2em -2\left( r-1\right) f+\left(2 r-1\right) d-\lambda \hfill \end{array}\right] $$

where

$$ \begin{array}{lll} a={\scriptscriptstyle \frac{\mu \left(1-\mu \right) w\left(3 c- w\right)}{{\left[ w\left(\mu +\alpha -1\right)- c\left(3\mu -1\right)\right]}^2}}\hfill & b={\scriptscriptstyle \frac{2{\mu}^2\left(\mu -1\right) w\left(3 c- w\right)\left(2 c-\alpha w\right)}{{\left[ w\left(\mu +\alpha -1\right)- c\left(3\mu -1\right)\right]}^3}}\hfill & {s}^{\ast }=-{\scriptscriptstyle \frac{{\left(\tilde{q}-1\right)}^2\left(2\tilde{q}-2{\tilde{q}}^2\right)}{{\left[1-\left(2\tilde{q}-2{\tilde{q}}^2\right)\right]}^2}}\hfill \\ {} f={\scriptscriptstyle \frac{\left[\left(\alpha -1\right) w+ c\right]\left(\alpha w-2 c\right)}{{\left[ w\left(\alpha +\mu -1\right)- c\left(3\mu -1\right)\right]}^2}}\hfill & d={\scriptscriptstyle \frac{2\mu \left[\left(\alpha -1\right) w+ c\right]{\left(\alpha w-2 c\right)}^2}{{\left[ w\left(\alpha +\mu -1\right)- c\left(3\mu -1\right)\right]}^3}}\hfill & {r}^{*}=-{\scriptscriptstyle \frac{\left({\tilde{p}}^2-2\tilde{p}\left)\right(2\tilde{p}-2{\tilde{p}}^2\right)}{{\left[1-\left(2\tilde{p}-2{\tilde{p}}^2\right)\right]}^2}}\hfill \end{array} $$

For S = (0, 0)

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{l}-2 a+ b-\lambda \kern2em -2 f+ d\hfill \\ {}\kern1em 2 a- b\kern3em 2 f- d-\lambda \hfill \end{array}\right] $$

which gives

$$ \lambda =0 $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−
0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−	0.00
0.10	0.00	0.00	0.00	0.00	0.00	0.00	−	0.00	0.00
0.25	0.00	0.00	0.00	0.00	0.00	−	0.00	0.00	0.00
0.50	0.00	0.00	0.00	0.00	−	0.00	0.00	0.00	0.00
0.75	0.00	0.00	0.00	−	0.00	0.00	0.00	0.00	0.00
0.90	0.00	0.00	−	0.00	0.00	0.00	0.00	0.00	0.00
0.99	0.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

$$ \lambda =-2 a+ b+2 f- d $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	+0.04	+0.04	+0.09	+1.17	+0.29	+0.17	−0.07	−923.62	−
0.01	+0.04	+0.04	+0.09	+1.16	+0.29	+0.18	−0.25	−	+1,057.52
0.10	+0.04	+0.04	+0.08	+1.03	+0.26	+0.17	−	+1.81	+1.37
0.25	+0.03	+0.03	+0.06	+0.62	+0.13	−	+0.27	+0.09	+0.08
0.50	+0.00	−0.00	−0.05	−2.69	−	−0.10	−0.01	−0.00	+0.00
0.75	−0.08	−0.10	−0.82	−	−1.15	−0.03	−0.01	−0.00	−0.00
0.90	−0.32	−0.47	−	−6.59	−0.35	−0.02	−0.00	−0.00	−0.00
0.99	−3.92	−	−1.00	−2.17	−0.17	−0.01	−0.00	−0.00	−0.00
1.00	–	−4.21	−0.73	−1.92	−0.16	−0.01	−0.00	−0.00	+0.00

For S = (0, 1)

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{l}-2 a+ b-\lambda \kern1em -2 f+ d\hfill \\ {}\kern2em b\kern4em d-\lambda \hfill \end{array}\right] $$

which gives

$$ \lambda ={\scriptscriptstyle \frac{1}{2}}\left[-\sqrt{-8\left( bf- ad\right)+{\left(2 a- b- d\right)}^2}-2 a+ b+ d\right] $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	−0.00	−0.01	−0.57	−0.16	−0.07	+0.01	+18.64	−
0.01	0.00	−0.00	−0.01	−0.58	−0.16	−0.07	+0.05	−	−960.50
0.10	0.00	−0.00	−0.01	−0.66	−0.22	−0.11	−	−0.89	−0.65
0.25	0.00	−0.00	−0.01	−0.85	−0.41	−	−0.20	−0.02	−0.02
0.50	0.00	−0.00	−0.05	−2.69	−	−0.10	−0.01	−0.00	0.00
0.75	0.00	−0.01	−0.36	−	−0.82	−0.02	−0.00	−0.00	0.00
0.90	0.00	−0.05	−	−4.27	−0.27	−0.01	−0.00	−0.00	0.00
0.99	0.00	−	−0.90	−2.07	−0.17	−0.01	−0.00	−0.00	0.00
1.00	–	−4.21	−0.73	−1.92	−0.16	−0.01	−0.00	−0.00	0.00

$$ \lambda ={\scriptscriptstyle \frac{1}{2}}\left[+\sqrt{-8\left( bf- ad\right)+{\left(2 a- b- d\right)}^2}-2 a+ b+ d\right] $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	−0.00	−0.00	−0.00	+0.32	+0.23	+1.46	+1,026.52	−
0.01	0.00	−0.00	−0.00	−0.00	+0.32	+0.24	+1.90	−	0.00
0.10	0.00	−0.00	−0.00	−0.00	+0.41	+0.64	−	−0.05	0.00
0.25	0.00	−0.00	−0.01	−0.07	+0.73	−	−0.00	−0.01	0.00
0.50	0.00	−0.00	−0.00	−0.00	−	+0.00	−0.00	+0.00	0.00
0.75	0.00	+0.00	0.04	−	+0.12	+0.00	+0.00	+0.00	+0.00
0.90	0.00	+0.00	−	−0.66	+0.01	+0.00	+0.00	+0.00	+0.00
0.99	0.00	−	−0.00	−0.00	+0.00	+0.00	+0.00	+0.00	+0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

For S = (1, 0)

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{l}- b-\lambda \kern2em - d\hfill \\ {}2 a- b\kern1em 2 f- d-\lambda \hfill \end{array}\right] $$

which gives

$$ \lambda ={\scriptscriptstyle \frac{1}{2}}\left[-\sqrt{-8\left( ad- b f\right)+{\left( b-2 f+ d\right)}^2}- b+2 f- d\right] $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	−0.00	−0.00	+0.00	−0.00	+0.00	−0.21	−943.11	−
0.01	0.00	−0.00	+0.00	−0.00	0.00	0.00	−0.45	−	+0.00
0.10	0.00	0.00	0.00	−0.00	+0.00	0.00	−	0.00	−0.00
0.25	0.00	−0.00	+0.00	−0.00	0.00	−	+0.00	+0.00	−0.00
0.50	0.00	0.00	0.00	0.00	−	0.00	0.00	0.00	0.00
0.75	−0.08	−0.09	−0.46	−	−0.26	−0.01	−0.00	−0.00	−0.00
0.90	−0.32	−0.42	−	−2.68	−0.07	−0.00	−0.00	−0.00	−0.00
0.99	−3.92	−	−0.10	−0.13	−0.01	−0.00	−0.00	−0.00	−0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

$$ \lambda ={\scriptscriptstyle \frac{1}{2}}\left[+\sqrt{-8\left( ad- b f\right)+{\left( b-2 f+ d\right)}^2}- b+2 f- d\right] $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	+0.04	+0.04	+0.10	+1.50	+0.32	+0.18	+0.00	0.00	−
0.01	+0.04	+0.04	+0.10	+1.50	+0.32	+0.18	0.00	−	+1,057.52
0.10	+0.04	+0.04	+0.10	+1.47	+0.28	+0.08	−	+1.85	+1.37
0.25	+0.03	+0.03	+0.08	+1.39	0.00	−	+0.38	+0.09	+0.08
0.50	0.00	+0.00	+0.00	+0.77	−	+0.02	+0.00	+0.00	−
0.75	0.00	+0.00	0.00	−	0.00	−0.00	−0.00	−0.00	−0.00
0.90	0.00	−0.00	−	+0.00	+0.00	−0.00	+0.00	−0.00	+0.00
0.99	0.00	−	+0.00	−0.00	+0.00	+0.00	+0.00	+0.00	+0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

For S = (1, 1)

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{l}- b-\lambda \kern2em - d\hfill \\ {}\kern1em b\kern2em d-\lambda \hfill \end{array}\right] $$

which gives

$$ \lambda =0 $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−
0.01	0.00	0.00	0.00	0.00	0.00	0.00	0.00	−	0.00
0.10	0.00	0.00	0.00	0.00	0.00	0.00	−	0.00	0.00
0.25	0.00	0.00	0.00	0.00	0.00	−	0.00	0.00	0.00
0.50	0.00	0.00	0.00	0.00	−	0.00	0.00	0.00	0.00
0.75	0.00	0.00	0.00	−	0.00	0.00	0.00	0.00	0.00
0.90	0.00	0.00	−	0.00	0.00	0.00	0.00	0.00	0.00
0.99	0.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

$$ \lambda =- b+ d $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	0.00	+0.00	+0.00	+0.25	−0.19	−0.16	−1.33	−1,025.67	−
0.01	0.00	+0.00	+0.00	+0.24	−0.19	−0.17	−1.75	−	+960.50
0.10	0.00	+0.00	+0.00	+0.22	−0.21	−0.44	−	+0.91	+0.65
0.25	0.00	+0.00	+0.00	+0.14	−0.19	−	+0.09	+0.02	+0.02
0.50	0.00	−0.00	−0.00	−0.77	−	−0.02	−0.00	−0.00	0.00
0.75	0.00	−0.00	−0.04	−	−0.19	−0.00	−0.00	−0.00	−0.00
0.90	0.00	−0.00	−	+1.01	−0.02	−0.00	−0.00	−0.00	−0.00
0.99	0.00	−	+0.00	+0.02	−0.00	−0.00	−0.00	−0.00	−0.00
1.00	−	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

For S = (s ^∗, r ^∗)

$$ J\left(\alpha, \mu \right)=\left[\begin{array}{cc}\hfill 2\left({s}^{\ast }-1\right) a-\left(2{s}^{\ast }-1\right) b-\lambda \hfill & \hfill 2\left({s}^{\ast }-1\right) f-\left(2{s}^{\ast }-1\right) d\hfill \\ {}\hfill -2\left({r}^{\ast }-1\right) a+\left(2{r}^{\ast }-1\right) b\hfill & \hfill -2\left({r}^{\ast }-1\right) f+\left(2{r}^{\ast }-1\right) d-\lambda \hfill \end{array}\right] $$

which gives

$$ \lambda =-{\scriptscriptstyle \frac{1}{2}}\left[2\left({r}^{\ast }-1\right) f-2\left({r}^{\ast }-1\right) d-2\left({s}^{\ast }-1\right) a+\left(2{s}^{\ast }-1\right) b\right]-\sqrt{\begin{array}{c}4\left({s}^{*}-1\right)\left({r}^{*}-1\right) a f-2\left({s}^{*}-1\right)\left(2{r}^{*}-1\right) a d\\ {}-2\left(2{s}^{*}-1\right)\left({r}^{*}-1\right) b f+\left(2{s}^{*}-1\right)\left(2{r}^{*}-1\right) b d\\ {}+\left[\left(2{r}^{*}-1\right) b-2\left({r}^{*}-1\right) a\right]\left[\;2\left({s}^{*}-1\right) f-\left(2{s}^{*}-1\right) d\right]\\ {}+{\scriptscriptstyle \frac{1}{4}}{\left[2\left({r}^{*}-1\right) f-2\left({r}^{*}-1\right) d-2\left({s}^{*}-1\right) a+\left(2{s}^{*}-1\right) b\right]}^2\end{array}} $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	+0.02	−	−	−2.94	−	−	−	−933.66	−
0.01	+0.02	−	−	−2.87	−	−	−	−	1,029.80
0.10	+0.02	−	−	−2.26	−	−	−	−	1.27
0.25	+0.01	−	−	−	−	−	−	−	−
0.50	0.00	−0.00	−0.02	−0.24	−	−0.06	−0.01	−0.00	0.00
0.75	−0.04	−0.04	−0.20	−	−	−0.01	−	−	−
0.90	−	−0.16	−	−	−	−0.01	−	−	−
0.99	−	−	−0.35	−	−	−0.00	−	−	−
1.00	−	−2.10	−0.37	−	−	−	−	−	−

$$ \lambda =-{\scriptscriptstyle \frac{1}{2}}\left[2\left({r}^{*}-1\right) f-2\left({r}^{*}-1\right) d-2\left({s}^{*}-1\right) a+\left(2{s}^{*}-1\right) b\right]+\sqrt{\begin{array}{c}4\left({s}^{*}-1\right)\left({r}^{*}-1\right) a f-2\left({s}^{*}-1\right)\left(2{r}^{*}-1\right) a d\\ {}-2\left(2{s}^{*}-1\right)\left({r}^{*}-1\right) b f+\left(2{s}^{*}-1\right)\left(2{r}^{*}-1\right) b d\\ {}+\left[\left(2{r}^{*}-1\right) b-2\left({r}^{*}-1\right) a\right]\left[2\left({s}^{*}-1\right) f-\left(2{s}^{*}-1\right) d\right]\\ {}+{\scriptscriptstyle \frac{1}{4}}{\left[2\left({r}^{*}-1\right) f-2\left({r}^{*}-1\right) d-2\left({s}^{*}-1\right) a+\left(2{s}^{*}-1\right) b\right]}^2\end{array}} $$

α|μ	0.00	0.01	0.10	0.25	0.50	0.75	0.90	0.99	1.00
0.00	+0.02	−	−	−3.31	−	−	−	−1,006.19	−
0.01	+0.02	−	−	−3.24	−	−	−	−	1,029.80
0.10	+0.02	−	−	−2.64	−	−	−	−	1.27
0.25	+0.01	−	−	−	−	−	−	−	−
0.50	−	−0.00	−0.02	−0.24	−	−0.01	−0.01	−0.01	0.00
0.75	−0.04	−0.06	−0.59	−	−	−	−	−	−
0.90	−	−0.32	−	−	−	−	−	−	−
0.99	−	−	−0.65	−	−	−	−	−	−
1.00	−	−2.10	−0.37	−	−	−	−	−	−