繁殖干渉アリー効果環境変動を考慮した有性生殖種 Titleと単為生殖種の種間競争の解析 ( 生物現象に対するモデリングの数理 ) Author(s) 岸, 茂樹 ; 深澤, 恵介 ; 大野, 理恵 Citation 数理解析研究所講究録 (2009), 1653: Issue Dat

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52-68 Issue Date 2009-06 URL http://hdl.handle.

1 繁殖干渉アリー効果環境変動を考慮した有性生殖種 Titleと単為生殖種の種間競争の解析 ( 生物現象に対するモデリングの数理 ) Author(s) 岸, 茂樹 ; 深澤, 恵介 ; 大野, 理恵 Citation 数理解析研究所講究録 (2009), 1653: Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

1653 2009 52-68 52 A mathematical model for interspecific competition between sexual and parthenogenetic species with reproductive interference, Allee effect and environmental fluctuation.

2 A mathematical model for interspecific competition between sexual and parthenogenetic species with reproductive interference, Allee effect and environmental fluctuation , 2, 3 1Shigeki Kishi, 2Keisuke Fukazawa and 3Rle Ohno 1Centerfor Ecological Research, Kyoto University, Otsu ciry, , JAPAN 2Graduate School ofmathematical Sciences, University oftokyo, Tokyo, , JAPAN 3Department of Mathematics, Graduate School of Sciences, Osaka University, Osaka city, , JAPAN Corresponding: kishi@ecology.kyoto-u.ac.jp Summary: In evolutionary biology, one of the most enigmatic problems is that parthenogenetic species are fairly rarer than sexual species in animal taxa, albeit population growth rate of parthenogenetic species can be twice as higher than sexual species which have to produce males as equal to females in reproductive investment, compared to parthenogenesis that does not need males. Some biologists argued that parthenogenesis should owe the two-fold cost of sex, which can cancel the benefit of the double growth rate. However, the cost on parthenogenesis has not been fully explained, althoug reduction of genetic variation and accumulation of deleterious genes have been suggested as the cost. Reproductive interference is an interspecific sexual interaction that adversely affects frtness of females through sexual indiscrimination between conspecifics and heterospecifics on reproductive process. Some mathematical studies predict that positive frequency dependence of reproductive interference is likely to drive interspecific competition to species exclusion between closely related species, while density dependence of resource competition is likely to bring coexistence. In the competition between sexual and the related parthenogenetic species, reproductive interference occurs in one-way from males of sexual species to females of parthenogenetic species. Based on Lotka-Volterra competition model, we built a simple mathematical competition model incorporating reproductive interference. To make the mathematical model more namral, we also assumed environmental flucmation and Allee effect, which reduces the population growth rate of sexual species with low density. Intense reproductive interference drove the competition to sexual exclusion of parthenogenetic species. However, even under intense reproductive interference, Allee effect allowed parthenogenetic species to survive when the competition started with the lower densities, contrast to the survival of sexual species when the competition started with the higher densities. Then, when environmental fluctuation was enoug large to cause extinctions of two species, parthenogenetic species repeatedly and dominantly occurred. We suggest that parthenogenesis is in a dilemma. Too high growth rate is likely to bring self-destruction together with environmental fluctuation, though too low growth rate is likely to result in sexual exclusion in terms of reproductive interference.

53 1 lntroduction 2 ( ) 80% 2% (Bell 1982) 2 Maynard-Smith (1978) 2 (the two-fold cost of sex) 2 2 1 2 2 (reviewed by Doncaster et al. 2000) 2 (Doncaster et al.

3 53 1 lntroduction 2 ( ) 80% 2% (Bell 1982) 2 Maynard-Smith (1978) 2 (the two-fold cost of sex) (reviewed by Doncaster et al. 2000) 2 (Doncaster et al. 2000) ( 2005) (Bell l982) ( 2005) Sphyrno tiburo (Chapman et al. 2007) Varanus komodoensis (Watts et al. 2006) (Bull 1991, Kuno 1992, reviewed by $Gr6ning$ and Hochkirch 2008)

4 54 (Volkov et al. 2005) (Allee effect) (Cuellar1977, Keamey2005) (Keamey2005) Model 1 2 1

$55.. 1 $N$ $b$ $h$ $d(b>d>0)$ $\frac{dn}{dt}=(b-d)n-hn^{2}$ 1 2 1 2 1 2 $c_{21\text{ }}$ 2 1 $c_{12}$ $N_{1\text{ }}$ $1$ 2 N2$

5 $N$ $b$ $h$ $d(b>d>0)$ $\frac{dn}{dt}=(b-d)n-hn^{2}$ $c_{21\text{ }}$ 2 1 $c_{12}$ $N_{1\text{ }}$ $1$ 2 N2 $\frac{dn_{1}}{dt}=(b_{1}-d_{1})n_{1}-h_{1}n_{1}(n_{1}+c_{12}n_{2})$ $\frac{dn_{2}}{dt}=(b_{2}-d_{2})n_{2}-h_{2}n_{2}(n_{2}+c_{21}n_{1})$ $c_{12}$

6 56 $c_{12}=1$ 1 2 $\frac{dn_{2}}{dt}=((\frac{n_{2}}{n_{2}+i_{21}n_{1}})b_{2}-d_{2})n_{2}-h_{2}n_{2}(n_{2}+c_{21}n_{1})$ $i_{21}$ $A(N_{1})= \frac{an_{1}}{1+an}1$ ( 2) $N$. $N$ $(a=1)$. $N_{1}\geq 0$ $0\leq A(N_{1})<1$ a $(\geq 0)$ a $\frac{dn_{1}}{dt}=(a(n_{1})b_{1}-d_{1})n_{1}-h_{1}n_{1}(n_{1}+c_{12}n_{2})$

7 57 $\frac{dn_{2}}{dt}=((\frac{n_{2}}{n_{2}+a(n_{1})i_{21}n_{1}})b_{2}-d_{2})n_{2}-h_{2}n_{2}(n_{2}+c_{21}n_{1})$ $R$ $(0\leq R\leq 1)$ $d$ $\frac{dn_{1}}{dt}=(a(n_{1})b_{1}-d_{1}(2\beta+1-f))n_{1}-h_{1}n_{1}(n_{1}+c_{12}n_{2})$ $\frac{dn_{2}}{dt}=((\frac{n_{2}}{n_{2}+a(n_{1})i_{21}n_{1}})b_{2}-d_{2}(2\beta+1-f))n_{2}-h_{2}n_{2}(n_{2}+c_{21}n_{1})$ $f$ $(0\leq f\leq 1)$ $f=1$ $2R+1- f=2r$ $0\leq 2R\leq 2$ $N_{1,t+s}=N_{1,t}+s \frac{dn_{1,t}}{dt}$ $N_{2,t+s}=N_{2,t}+s \frac{dn_{2,t}}{dt}$ $s$ $t$ $(0\leq s\leq 1)$ $s=1$ $s=1$ $R$ 1 2 $R$ $N_{1\text{ }}$ N2 $(N, \geq 0, N_{2}\geq 0)$ 0.01 $N_{1.t+s},$ $N_{2.t+s}$ (zero-growth isocline)

8 58 3 Results $2(b_{1}- d_{1})=b_{2}- d_{2}$ 2 c $==$ C $=0.5$ $a=1$ Reproductive interference 1 2 ( 3) 3. $ _{21}=0.1$ $ _{21}=0.5$ $N$ $N$ 25 $50_{f}$ $f$ 3. 1( ) 2( ). 2 $(i_{21}=0.1)$ 2 $(i_{21}=0.5)$. 2 $(N_{1,0}=N_{2,0}=1)$ 2 : $N_{1,0}=5$, N2,0 1 : $=$ $N_{1,0}=1,$ $N_{2,0}=5)$. $(b_{1}=1.5,$ $b_{2}=$ $2,$ $d_{1}=d_{2}=1,$ $h_{1}=h_{2}=0.01)$ ( 3) ( 3 ) 1 2 ( 3 ) ( 3 ) 2 1

9 ( 3 ) ( 3 ) 1 ( 3 $)$ Reproductive $interference+allee$ effect $4a, c)$ 4. $N_{1,0}=N_{2_{l}0}=5$ $N_{1,0}=N_{2,0}=10$ $N_{1},0^{arrow}arrow N_{2,0}=5$ $c =0.1$ 4. 1( ) 2( ). $a,$ $c,$ $e)$ $b,$ $d,$ $f)$. $(i_{21}=0.1;a, b)$ $(i_{21}=0.5;c, d)$ $(i_{21}=0.5, c =0.1;e, f)$. $a,$ $b,$ $e,$ $f$ $N_{1,0}=N_{2,0}=5$ $c,$ $d$ $N_{1,0}=N_{2.0}=10$. $(b_{1}=1.5, b_{2}=2, d_{1}=d_{2}=1, h_{1}=h_{2}=0.01)$ 2 2 $4a)$

$60 1 $4_{C})$ 2 2 ( 1 $4b)$ 2 $4c, d)$ $4a)$ 1 $4e, f)_{0}$ Reproductive $interference+alleeeffect+environmental$ Fluctuation 1 2 5. 5. 1( ) 2( ). $(c =0.2, i_{21}=0.2)$ ( 5a) $5b)$ $5c)$.$

10 60 1 $4_{C})$ 2 2 ( 1 $4b)$ 2 $4c, d)$ $4a)$ 1 $4e, f)_{0}$ Reproductive $interference+alleeeffect+environmental$ Fluctuation ( ) 2( ). $(c =0.2, i_{21}=0.2)$ ( 5a) $5b)$ $5c)$. $5d)$ $5e)$ 1 ( $5f)$. $b_{1}=1.5,$ $b_{2}=2,$ $d_{1}=d_{2}=$ $1,$ $h_{1}=h_{2}=0.01,$ $N_{1.0}=N_{2,0}=5$.. $(c =0.2, i21=0.2)$ $(f=0)2$ $5a)$ 1 2 $5b)$ $5c)$ 2 1 ( $5d)$ 1 $5e)$ 1 2 $5f)$

11 61 $(i_{21}=0.5)$ 1 $6a, b)$ $6a, c, e)$ $6b, df)$ (i21 0.5) ( $=$ $6a,$ $b)$ $6c, d)$ $6e, f)$ $b_{1}=1.5,$ $b_{2}=2,$ $d_{1}=d_{2}=$ 1, $h_{1}=h_{2}=0.01,$ $N_{1,0}=N_{2,0}=5$. 2 $6a)$ 2 $6b)$ 2 $6c, d)$ 0.01 $6c, d)$ 2 $6c)$ 1 $6d)$ $6a, e)$ 1

12 62 $6a)$ 2 $6e)$ $6e, f)$ 2 ( 6e) 1 $6f)$ ( 2) ( 7) (a) $(b)$ (c). $1h$ $b_{1}=1.5,$ $b_{2}=2,$ $d_{1}=d_{2}=1,$ $h_{1}=h_{2}=0.01$, c $=0.2,$ $i_{21}=0.5,$ $N_{1,0}=N_{2,0}=5$. $7a)$ ( $7b)$ $7c)$ Isocline analysis ( 8)

13 $ $ $N_{1}$ ( ) 2( ).. $dn_{1}/dt=0,$ $dn_{2}/dt=0$ 1 2 (Nl, N2). (a) a $=0$ $a=1)$ (b) (c) $(i_{21}=0.5)$ (d) 2 $(b_{1}=2, b_{2}=3, i_{21}=0.5)$. $b_{1}=1.5,$ $b_{2}=2,$ $d_{1}=d_{2}=1,$ $h_{1}=h_{2}=0.01,$ $c =0.2,$ $i_{21}=0.2$ ( $8a)$ $8b)$ 1 $8c)$ $8c)$ 2 3 $8d)$

14 64 4 Discussion $1)_{\text{ }}$

$frenatus Hgarnotti (D\ {s}ame&petren 2006) Ramphotyphlops braminus Lepidodactylus lugubris Hemiphyllodactylus typus typus ( 2000) Cyprinidae Carassius auratus langsdorfii (Hakoyama and Iguchi 2002)$

15 65 1. ( 1) Hemidaciylus gamotti (Case et al. 1994) Hfrenatus (Case et al. 1994) $H$ garnotti $H$. frenatus Hgarnotti (D\ {s}ame&petren 2006) Ramphotyphlops braminus Lepidodactylus lugubris Hemiphyllodactylus typus typus ( 2000) Cyprinidae Carassius auratus langsdorfii (Hakoyama and Iguchi 2002) Cauratus Ccuvieri (Hakoyama and Iguchi2002) 1976 Lissorhoptms oryzophilus Oryza sativa (Saito et al.2005) (Saito et Aphidoidea Tetranichidae (Simon et al. 2003) (Hoffinann et al. 2008) Homo sapiens Plasmodium 4 3 Pvivax $4$ Pmalariae Pfalcipamm Anqpheles

$2008) (2008) Tjaponicum 2 (Takakura et al.2008) (Takakura et al. 2008) 3 2 $i_{21}=0.5$ $i_{21}=0.5$ 1:1 1/3 (Kishi et al.$

16 66 (Paul et al. 2003) rcacum albidum 3 (Morita et al. 1990) T. q cinale (Collier and Rogstad2004) (Takakura et al. 2008) (2008) Tjaponicum 2 (Takakura et al.2008) (Takakura et al. 2008) 3 2 $i_{21}=0.5$ $i_{21}=0.5$ 1:1 1/3 (Kishi et al. in press)

17 67 2 Acknowledgements $2008$ ( ) References Bell, $G$ The masterpiece of nature: the evolution and genetics of sexuality. University of Califomia Press, Berkeley, USA. Bull, C. M Ecology ofparapatric distributions. Annual Review of Ecology and Systematics 22: Case, T. J., Bolger, D. T. and Petren, K Invasions and competitive displacement among house geckos in the tropical Pacific. Ecology $75:464\triangleleft 77$. Chapman, D. D., Shivji, M. S., Louis, Ed., Sommer, J., Fletcher, H. and $Prod6hl$, P. A Virgin birth in a hammerhead shark. Biology Letters 4: Collier, M. H. and Rogstad, S. H Clonal variation in floral stage timuing in the common dandelion Taraxacum officinale (Asteraceae). Americal Joumal ofbotany 91: Dame, E. A. and Petren, K Behavioral mechanism of invasion and displacement in Pacific island geckos (Hemidactylus). Aminal Behaviour 71: Doncaster, C. P., Pound, G E. and Cox, J. C The ecological cost of sex. Nature 404: $Gr6ning$, J. and Hochkirch, A Reproductive interference between animal species. Quarterly Review of Biology 83: Hakoyama, H. and Iguchi, K Male mate choice in the gynogenetic-sexual complex of cmcian carp, Carassius auratus. Acta Ethologica 4:85-90.

18 68 Hoffmann, A. A., Reynolds, K. T., Nash, M. A. and Weeks, A. R A high incidence of parthenogenesis in agricultural pests. Proceedings of the Royal Society $B$ 275: Kuno, E Competitive exclusion through reproductive interference. Researches on Population Ecology 34: Maynard-Smith, J The evolution of sex. Cambridge University Press, Cambridge, UK : Morita, T., Sterk, A. and den Nus, J. C. M The significance of agamospermous triploid pollen donors in the sexual relationships between diploids and triploids in Taraxacum (Compositae). Plant Sciences Biology 5: No. 345, pp Paul, R. E. L., Ariey, F. and Robert, V The evolutionary ecology ofplasmodium. Ecology Letters 6: Saito, T., Hirai, K. and Way, M. O The rice water weevil, Lissorhoptrus oryzophilus Kuschel (Coleoptera: Curculionidae). Applied Entomology and Zoology 40: Simon, J. C., Delmotte, F., Rispe, C. and Crease, T Phylogenetic relationships between parthenogens and their sexual relatives: the possible routes to parthenogenesis in animals. Biological Journal ofthe Linnean Society 79: Volkov, I., Banavar, J.R., He, F., Hubbel, S.P. and Marita4 A Density dependence explains tree species abundance and diversity in tropical forests. Nature 438: Takakura, K. I., Nishida, T., Matsumoto, T. and Nishida, S Alien dandelion reduces the seed set of a native congener throug frequency dependent and one sided effects. Biological Invasions (online first). Watts, P. C., Buley, K. R., Sanderson, S., Boardman, W., Ciofi, C. and Gibson, R Parthenogenesis in komodo dragons. Nature 444: $1021arrow 1022$.

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繁殖干渉 アリー効果 環境変動を考慮した有性生殖種 Titleと単為生殖種の種間競争の解析 ( 生物現象に対するモデリングの数理 ) Author(s) 岸, 茂樹 ; 深澤, 恵介 ; 大野, 理恵 Citation 数理解析研究所講究録 (2009), 1653: Issue Dat

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Title 泌尿器科領域に於ける17-Ketosteroidの研究 17-Ketosteroidの臨床的研究 第 III 篇 : 尿 Author(s) 卜部, 敏入 Citation 泌尿器科紀要 (1958), 4(1): 3-31 Issue Date URL

繁殖干渉アリー効果環境変動を考慮した有性生殖種 Titleと単為生殖種の種間競争の解析 ( 生物現象に対するモデリングの数理 ) Author(s) 岸, 茂樹 ; 深澤, 恵介 ; 大野, 理恵 Citation 数理解析研究所講究録 (2009), 1653: Issue Dat

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Title 泌尿器科領域に於ける17-Ketosteroidの研究 17-Ketosteroidの臨床的研究第 III 篇 : 尿 Author(s) 卜部, 敏入 Citation 泌尿器科紀要 (1958), 4(1): 3-31 Issue Date URL