General Form: Lotka–Volterra Predator-Prey

\[{{d}\over{d\,t}}\,N_{i}= b_{i} N_{i}\left[C S\right]_{i} - d_{i} N_{i} \notag\]

\[{{d}\over{d\,t}}\,S_{j}=a_{j} S_{j}\,\left(1-{{S_{j}}\over{k_{j}}} \right)- S_{j} \left[{\it C^T} N\right]_{j} \notag\], where \(C\) is an \(n \, by \, n\) consumption matrix


Predator

  • \(N_i\) : Population size of Predator \(i\).    \(i=1: Sex\),    \(i \neq 1: Asex\).

  • \(b_i\) : Birth rate of Predator \(i\)

  • \(d_i\) : Death rate of Predator \(i\)

  • \(C\) : Records the proportion of each Prey \(j\) comsumed by each Predator \(i\) (niche breadths).

    \(\hspace{0.5 mm}\sum_{j=1}^{n} c_{1j} = \sum_{j=1}^{n} c_{2j} = \hspace{1.0 mm} ... \hspace{1.0 mm} = \sum_{j=1}^{n} c_{nj} = row \, sum\).


Prey

  • \(S_j\) : Population size of Prey \(j\)

  • \(a_j\) : Intrinsic growth rate of Prey \(j\)

  • \(k_j\) : Carrying capacity of Prey \(j\)



2-Predator & 2-Prey

\[{{d}\over{d\,t}}\,N_{1} =\left[ b_{1} N_{1}\,\left( S_{2}\,c_{12}+S_{1}\,c_{11}\right)- d_{1} N_{1}\, \right] \label{eq:N1}\]

\[{{d}\over{d\,t}}\, N_{2} =\left[ b_{2} N_{2}\,\left( S_{2}\,c_{22}+S_{1}\,c_{21}\right)- d_{2} N_{2}\, \right] \label{eq:N2}\]

\[{{d}\over{d\,t}}\, S_{1} =\left[ a_{1} S_{1}\,\,\left( 1-{{S_{1}}\over{k_{1}}}\right)-S_{1}\,\left(N_{2}\,c_{21}+N_{1}\,c_{ 11}\right) \right] \label{eq:s1}\]

\[{{d}\over{d\,t}}\, S_{2} =\left[ a_{2} S_{2}\,\,\left( 1-{{S_{2}}\over{k_{2}}}\right)-S_{2}\,\left(N_{2}\,c_{22}+N_{1}\,c_{ 12}\right) \right] \label{eq:s2}\]

Conversion to Lotka-Volterra Competition Equations

  1. Set eq. \(\eqref{eq:s1}\) & \(\eqref{eq:s2}\) to \(0\) and solve for \(equilibrium \, \, S_j^*\)

\[{\it S_1^*} = -{{\left({\it N_2} {\it c_{21}} + {\it N_1} {\it c_{11}} - {\it a_1}\right) {\it k_1}}\over{{\it a_1}}}, \,\, {\it S_1^*}=0 \label{eq:s1e}\]

\[{\it S_2^*}=-{{\left({\it N_2} {\it c_{22}} + {\it N_1} {\it c_{12}}-{\it a_2}\right) {\it k_2}}\over{{\it a_2}}}, \,\, {\it S_2^*}=0 \label{eq:s2e}\]


  1. Replace \(S_j\) with \(S_j^*\) in eq. \(\eqref{eq:N1}\) & \(\eqref{eq:N2}\):

\[{{d}\over{d\,t}}\,{\it N_1}={\it N_1}\,{\it b_1}\,\left(-{{ {\it c_{12}}\,\left({\it N_2}\,{\it c_{22}}+{\it N_1}\, {\it c_{12}}-{\it a_2}\right)\,{\it k_2}}\over{{\it a_2}}}- {{{\it c_{11}}\,\left({\it N_2}\,{\it c_{21}}+{\it N_1}\, {\it c_{11}}-{\it a_1}\right)\,{\it k_1}}\over{{\it a_1}}} \right)-{\it N_1}\,{\it d_1} \label{eq:comp1}\]

\[{{d}\over{d\,t}}\,{\it N_2}={\it N_2}\,{\it b_2}\,\left(-{{ {\it c_{22}}\,\left({\it N_2}\,{\it c_{22}}+{\it N_1}\, {\it c_{12}}-{\it a_2}\right)\,{\it k_2}}\over{{\it a_2}}}- {{{\it c_{21}}\,\left({\it N_2}\,{\it c_{21}}+{\it N_1}\, {\it c_{11}}-{\it a_1}\right)\,{\it k_1}}\over{{\it a_1}}} \right)-{\it N_2}\,{\it d_2} \label{eq:comp2}\]



  1. Set Consumption Matrix, \(\pmatrix{c_{11}&c_{12}\cr c_{21}&c_{22}\cr }\) with \(c_{21}=0\), i.e. \(C = \pmatrix{c_{11}&c_{12}\cr 0&c_{22}\cr }\):

\[{{d}\over{d\,t}}\,{\it N_1}={\it N_1}\,{\it b_1}\,\left(-{{ {\it c_{12}}\,\left({\it N_2}\,{\it c_{22}}+{\it N_1}\, {\it c_{12}}-{\it a_2}\right)\,{\it k_2}}\over{{\it a_2}}}- {{{\it c_{11}}\,\left({\it N_2}\,{\it c_{21}}+{\it N_1}\, {\it c_{11}}-{\it a_1}\right)\,{\it k_1}}\over{{\it a_1}}} \right)-{\it N_1}\,{\it d_1} \label{eq:comp3}\]

\[{{d}\over{d\,t}}\,{\it N_2}=-{{{\it N_2}\,{\it b_2}\, {\it c_{22}}\,\left({\it N_2}\,{\it c_{22}}+{\it N_1}\, {\it c_{12}}-{\it a_2}\right)\,{\it k_2}}\over{{\it a_2}}}- {\it N_2}\,{\it d_2} \label{eq:comp4}\]

  1. Comparison to LK competition equations

\[{{d}\over{d\,t}}\,{\it N_1}={\it r_1}{\it N_1} \left(1-{{{\it N_1} + {\it N_2}{\it \alpha_{12}}}\over{{\it K_1}}}\right)\, - {\it d_1}{\it N_1}\]

\[{{d}\over{d\,t}}\,{\it N_2} = {\it r_2}{\it N_2} \left(1-{{{\it N_2} + {\it \alpha_{21}}{\it N_1}}\over{{\it K_2}}}\right) - {\it d_2} {\it N_2}\]


Asumptions

Consumption Matrix

\[C_{n \times n} = \pmatrix{ c_{11}&c_{12}&...&c_{1n}\cr c_{21}&c_{22}&...&c_{2n}\cr .& .& &.\cr .& .& &.\cr .& .& &.\cr c_{n1}&c_{n2}&...&c_{nn}\cr }, \,\,\,\, \sum_{j=1}^{n} c_{1j} = \sum_{j=1}^{n} c_{2j} = \,\, ... \,\, = \sum_{j=1}^{n} c_{nj} = row \, sum, \, c_{ij}>0 \label{eq:matr}\]

Sex

  • Cost: \(\bf b_1=0.5 \times b_i, \, i \neq 1\) in eq. \(\eqref{eq:comp3}\) & \(\eqref{eq:comp4}\)

  • Benefit: Broad niche (Generalist). Consumption matrix: \(\bf all \, c_{1j} \neq 0\)

Asexual Clones

  • Narrow niche (Specialist)

Example

\[C_{4 \times 4} = \pmatrix{ 0.25&0.25&0.25&0.25\cr 0.7&0.1&0.1&0.1\cr 0.1&0.7&0.1&0.1\cr 0.1&0.1&0.1&0.7\cr}\]

  • row 1: Sex

  • row 2~4: Asex

Results

Sex vs. 1 Asex: 2-Predator & 2-Prey

\(C = \pmatrix{c_{11}&c_{12}\cr 0&c_{22}\cr }\)


General Cases (More than 1 Asexual clones)

  • Sex may become extinct, outcompete or coexist with asexual clone(s)

  • Depend on niche breadths of each asexual clones, i.e. the setting of the consumption matrix, eq. \(\eqref{eq:matr}\)

    • Sex can exist if it retains at least a small resourse refuge