Competition and coexistence of two species for one nutrient with internal storage and predation Yi Hui Ho National Tsing-Hua University, Taiwan ( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.) Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 1 / 19
Models with variable yield: Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 2 / 19
R ( t ) = Concentration of nutrient at time t, N 1 , N 2 =The population density of microorganism, Q i ( t ) = The average amount of stored nutrient per cell of i - th population at time t, R (0) = input concentration of nutrient, D = dilution rate, µ i ( Q i ) : the per-capita growth rate of of species i , f i ( R , Q i ) : the per-capita uptake rate of of species i , Q min , i : the threshold cell quota below which no growth of species i occurs. Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 3 / 19
Assumptions We assume that µ i ( Q i ) is defined and continuously differentiable for Q i ≥ Q min , i > 0 and satisfies � µ i ( Q i ) ≥ 0 , µ ′ i ( Q i ) > 0 and is continuous for Q i ≥ Q min , i , ( H 1) µ i ( Q min , i ) = 0 . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 4 / 19
f i ( R , Q i ) is continuously differentiable for R > 0 and Q i ≥ Q min , i and satisfies � ∂ f i ∂ f i ( H 2) f i (0 , Q i ) = 0 , ∂ R > 0 , ∂ Q i ≤ 0 . In particular, f i ( R , Q i ) > 0 when R > 0. Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 5 / 19
Results: the competitive exclusion principle holds. (Smith Hal and Waltman, SIAM J. Appl. Math. 1994) Smith and Waltman proved only one of the populations survives by the method of monotone dynamical system. Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 6 / 19
The Model Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 7 / 19
N 1 ( t ) = The population density of the autotroph, N 2 ( t ) = The population density of the mixotroph, g ( N 1 ) : the functional response of the mixotroph feeding on the autotroph, g max N b 1 g ( N 1 ) = , b > 1 , Holling type III ( k max ) b + N b 1 g ( N 1 ) Q 1 : the assimilation of nutrients from ingested prey, The nutrient uptake rates f i ( R , Q i ) satisfies ( H 2) and the growth rate µ i ( Q i ) satisfies ( H 1). Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 8 / 19
The growth rate and uptake rate: 1 The growth rate µ i ( Q i ) takes the form: � 1 − Q min , i � µ i ( Q i ) = µ i ∞ , Q i where µ i ∞ is the maximal growth rate of species i , Q min , i is the threshold cell quota below which no growth of species i occurs. 2 The uptake rate f i ( R , Q i ) takes the form: R f i ( R , Q i ) = ρ max , i ( Q i ) K i + R , Q i − Q min , i ρ max , i ( Q i ) = ρ high max , i − ( ρ high max , i − ρ low max , i ) , Q max , i − Q min , i where Q min , i ≤ Q i ≤ Q max , i ( a maximal possible quota ) . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 9 / 19
Set W ( t ) = R (0) − R − Q 1 N 1 − Q 2 N 2 in (1 . 1) and note that dW dt = − DW . Then we can rewrite (1 . 1) as follows: dN 1 dt = [ µ 1 ( Q 1 ) − D ] N 1 − g ( N 1 ) N 2 , dt = f 1 ( R (0) − Q 1 N 1 − Q 2 N 2 − W , Q 1 ) − µ 1 ( Q 1 ) Q 1 , dQ 1 dN 2 dt = [ µ 2 ( Q 2 ) − D ] N 2 , dt = f 2 ( R (0) − Q 1 N 1 − Q 2 N 2 − W , Q 2 ) − µ 2 ( Q 2 ) Q 2 + g ( N 1 ) Q 1 , dQ 2 dW dt = − DW , N i (0) ≥ 0 , Q i (0) ≥ Q min , i , i = 1 , 2 , (2.1) Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 10 / 19
Putting W = 0 in (2 . 1), we arrive at the following reduced system of (1 . 1): dN 1 dt = [ µ 1 ( Q 1 ) − D ] N 1 − g ( N 1 ) N 2 , dt = f 1 ( R (0) − Q 1 N 1 − Q 2 N 2 , Q 1 ) − µ 1 ( Q 1 ) Q 1 , dQ 1 dN 2 (3.1) dt = [ µ 2 ( Q 2 ) − D ] N 2 , dt = f 2 ( R (0) − Q 1 N 1 − Q 2 N 2 , Q 2 ) − µ 2 ( Q 2 ) Q 2 + g ( N 1 ) Q 1 , dQ 2 N i (0) ≥ 0 , Q i (0) ≥ Q min , i , i = 1 , 2 , with initial values in the domain Σ = { ( N 1 , Q 1 , N 2 , Q 2 ) ∈ R 4 + : Q i ≥ Q min , i , Q 1 N 1 + Q 2 N 2 ≤ R (0) } . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 11 / 19
Equilibrium Analysis 1 E 0 = ( N 1 , Q 1 , N 2 , Q 2 ) = (0 , Q 0 1 , 0 , Q 0 2 ) : E 0 always exists, Q 0 i is the unique solution of f i ( R (0) , Q i ) − µ i ( Q i ) Q i = 0 , i = 1 , 2 . 2 E 1 = ( N ∗ 1 , Q ∗ 1 , 0 , Q ∗ 2 ) : µ 1 ( Q ∗ 1 ) = D f 1 ( R (0) − Q ∗ 1 N ∗ 1 , Q ∗ 1 ) = DQ ∗ 1 f 2 ( R (0) − Q ∗ 1 N ∗ 1 , Q ∗ 2 ) − µ 2 ( Q ∗ 2 ) Q ∗ 2 + g ( N ∗ 1 ) Q ∗ 1 = 0 . 3 E 2 = (0 , Q ∗∗ 1 , N ∗∗ 2 , Q ∗∗ 2 ) : µ 2 ( Q ∗∗ 2 ) = D f 2 ( R (0) − Q ∗∗ 2 N ∗∗ 2 , Q ∗∗ 2 ) = DQ ∗∗ 2 f 1 ( R (0) − Q ∗∗ 2 N ∗∗ 2 , Q ∗∗ 1 ) − µ 1 ( Q ∗∗ 1 ) Q ∗∗ 1 = 0 . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 12 / 19
E 0 = ( N 1 , Q 1 , N 2 , Q 2 ) = (0 , Q 0 1 , 0 , Q 0 2 ) E 1 = ( N ∗ 1 , Q ∗ 1 , 0 , Q ∗ 2 ) , E 2 = (0 , Q ∗∗ 1 , N ∗∗ 2 , Q ∗∗ 2 ) , Lemma The following statements are true: (i) E 0 is locally asymptotically stable if both µ i ( Q 0 i ) < D, i = 1 , 2 ; (ii) E 0 is unstable if µ i ( Q 0 i ) > D, for some i; (iii) E i exists if and only if µ i ( Q 0 i ) > D, i = 1 , 2 . Lemma Suppose that E 1 and E 2 exist. (i) E 1 is locally asymptotically stable if µ 2 ( Q ∗ 2 ) − D < 0 , and unstable if µ 2 ( Q ∗ 2 ) − D > 0 . (ii) E 2 is locally asymptotically stable if µ 1 ( Q ∗∗ 1 ) − D < 0 , and unstable if µ 1 ( Q ∗∗ 1 ) − D > 0 . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 13 / 19
We give the following assumptions: ( A0 ) E 1 and E 2 exist, that is, µ i ( Q 0 i ) > D , i = 1 , 2. ( A1 ) E 1 is unstable, that is, µ 2 ( Q ∗ 2 ) − D > 0. ( A2 ) E 2 is unstable, that is, µ 1 ( Q ∗∗ 1 ) − D > 0. Let Σ 0 = { ( N 1 , Q 1 , N 2 , Q 2 ) ∈ Σ : N 1 > 0 , N 2 > 0 } , ∂ Σ 0 := Σ \ Σ 0 . (1) Theorem Let ( A0 ) , ( A1 ) and ( A2 ) hold. Then system (3 . 1) is uniformly persistent with respect to (Σ 0 , ∂ Σ 0 ) in the sense that there is an η > 0 such that for any ( N 1 (0) , Q 1 (0) , N 2 (0) , Q 2 (0) ∈ Σ 0 , the solution ( N 1 ( t ) , Q 1 ( t ) , N 2 ( t ) , Q 2 ( t )) of (3 . 1) satisfies lim inf t →∞ N i ( t ) ≥ η, i = 1 , 2 . Further, system (3 . 1) admits at least one positive (coexistence) solution. Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 14 / 19
Theorem Let ( A0 ) , ( A1 ) and ( A2 ) hold. Then system (1 . 1) admits at least one positive (coexistence) solution, and there is an η > 0 such that for any initial value ( R (0) , N 1 (0) , Q 1 (0) , N 2 (0) , Q 2 (0)) ∈ Ω with N 1 (0) > 0 and N 2 (0) > 0 , the corresponding solution of (1 . 1) satisfies lim inf t →∞ N i ( t ) ≥ η, i = 1 , 2 . Yi Hui Ho (National Tsing-Hua University, Taiwan( Joint work with Drs. Sze-Bi Hsu and Feng-Bin Wang.)) Competition and coexistence of two species for one nutrient with internal storage and predation 15 / 19
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